# A Topological Representation of Semantics of First-order Logic and Its
Application as a Method in Model Theory

Yunfei Qin
Peking University
<EMAIL_ADDRESS>

###### Abstract

Various topological concepts are often involved in the research of
mathematical logic, and almost all of these concepts can be regarded as
developing from the Stone representation theorem. In the Stone representation
theorem, a Boolean algebra is represented as the algebra of the clopen sets of
a Stone space. And based on this, a natural connection is established between
the structure of Stone space and the semantics of propositional logic. In
other words, models of a propositional theory are represented as points in a
Stone space. This enables us to use the concepts of topology to describe many
facts in logic. In this paper, we do the same thing for the first-order logic.
That is, we organize the basic objects of semantics of first-order logic, such
as theories, models, elementary embeddings, and so on, into a kind of
topological structure defined abstractly. To be precise, this kind of
structure is a kind of enriched-topological space which we call cylindric
space in this paper. Furthermore, based on this topological representation of
semantics of first-order logic, we systematically introduce a method of point-
set topology into the research of model theory. We demonstrate the great
advantages of this topological method with an example and provide a general
discussion of its features, advantages, and connection to the type space.

_K_ eywords Topology $\cdot$ Stone space $\cdot$ Model theory $\cdot$
Cylindric algebra

##  1 Introduction

Logic has been closely related to topology for a long time, and the most
important connection is Stone duality. Stone duality is a further development
of the Stone representation theorem of Boolean algebra. In Stone’s
representation theorem, Boolean algebra is represented as a set field algebra
on the power set of its ultrafilters, and this set field algebra has a natural
topological structure. In Stone duality, this topological structure is
characterized as Stone space [10]. The connection between Stone duality and
logic is that Stone duality can be regarded as a syntax-semantic duality of
propositional logic: on the one hand, it is well known that the Lindenbaum-
Tarski algebra of the syntactic structure of propositional logic is a Boolean
algebra; On the other hand, since the Boolean homomorphisms from a Boolean
algebra to 0-1 algebra correspond to the ultrafilters on the Boolean algebra,
the points in the dual Stone space of this Boolean algebra can be regarded as
such Boolean homomorphisms, and these homomorphisms can be regarded as an
abstraction of the model of propositional logic theory (0-1 assignment to
propositional variables) [7]. Based on this, a Stone space can be regarded as
a topological space consisting of all the models of a propositional theory. In
this way, Stone duality establishes an algebraic-geometric duality between the
syntactic structure of propositional logic theory and its semantic
interpretation and gives us a systematic topological perspective on the
semantics of propositional logic.

As a generalization of propositional logic, first-order logic naturally
inherits the topological intuition . Various topological concepts are common
in the study of first-order model theory. Even in many early works, the
topology method has a place (two obvious examples can be seen in [6] and
[17]). Thus, a question arises: to what extent can we develop these
topological concepts and intuitions in first-order logic? Can we organize the
models of first-order theory into a certain topological structure, just as
organizing the models of propositional logic as a Stone space? This paper
gives a positive answer to these questions. That is, the development in this
direction is not only possible but also has considerable potential to be
applied as a systematic method to the further study of first-order model
theory.

First, we review the existing works on topologizing first-order logic.

Unlike propositional logic, a first-order structure has a natural way of
constructing a set field representation algebra of it, that is, cylindric set
algebra. Each of these algebras is a set field algebra obtained by adding some
non-Boolean operators to the set field Boolean algebra consisting of sequences
of length $\alpha$($\alpha$-sequence, in short) on a set. For a first-order
structure, consider the set $S$ consisting of all $\omega$-sequences on the
domain of this first-order structure. Each definable set of this structure can
naturally correspond to a subset of $S$, and all such subsets of $S$ form a
cylindric set algebra. Obviously, this algebra generated by the first-order
structure is a Boolean algebra and thus naturally has a zero-dimensional
topological structure.

Cylindric set algebras are related to cylindric algebra proposed by Henkin,
Monk, and Tarski in [9]. Chapter 1 of [8] mentions that the cylindric algebra
of local-finite dimension is the algebraic abstraction of Lindenbaum-Tarski
algebras of theories of the first-order language with equality and without
functions and constants. Generally speaking, some but not every cylindric
algebra can be represented as a cylindric set algebra. The Resek-Thompson
theorem tells us that the cylindric set algebra is a set field representation
of a modified version of cylindric algebra (see [1]). However, considering the
soundness and completeness of first-order logic, it is easy to see that each
cylindric set algebra generated by a first-order structure is the set field
representation of a local-finite dimensional cylindric algebra.

In addition to the topology on the cylindric set algebra of the first-order
structure mentioned above, there has been much research on constructing the
corresponding topological space of cylindric algebra in the following two
early works.

In [5], as a generalization of Stone duality, Comer Stephen dualized a
cylindric algebra to a class of sheaves derived from it, which are, of course,
based on topological space. However, he did not relate this work to the
semantics of first-order logic.

In [14], Charles Pinter investigated the Stone dual space of the underlying
Boolean algebra of the local-finite dimensional cylindric algebra (in fact, it
is not a standard cylindric algebra but an equivalent form of cylindric
algebra proposed in [16]). He showed that a cylindrification $c_{\kappa}$ of a
cylindric algebra could be dualized as the following equivalent relation
$\sim_{\kappa}$ on the dual space:

$p\sim_{\kappa}q\Leftrightarrow\\{c_{\kappa}a:a\in p\\}\subseteq q.$

On this basis, he showed how to construct a first-order structure based on an
ultrafilter with Henkin property of algebraic elements. However, his
discussion was limited to the case of countable-dimension, so that only
countable models can be constructed in this way. Moreover, the continuous
mappings between spaces were not mentioned.

There are also works related to the topologization of semantics of first-order
logic that does not involve cylindric algebra. Based on the results of Butz
and Moerdijk in [4], Awodey and Forssell constructed a first-order logical
duality based on category logic in [3]. They established a duality between the
Boolean coherent category representing the first-order theory and a kind of
topological groupoids, in which the dual topological groupoid of the Boolean
coherent category representing a theory $T$ can be regarded as consisting of
enough $T$-models. This duality, although providing a topological
representation of first-order structures, did not consider how to represent
the elementary embeddings between the structures, which are clearly an
important part of the semantics of first-order logic.

The work in this paper can be seen as a development of Pinter’s investigation
of the Stone dual space of cylindric algebra. However, this paper will not
deal with cylindric algebra in technical detail. It is easy to see that the
ultrafilters on the Lindenbaum algebra of a first-order theory are essentially
the algebraic counterpart of the maximal consistent sets in logic, so by
replacing the ultrafilters in the Stone dual space with the MCSs, we obtain a
space consisting of first-order logic objects, which we call model space in
this paper (For a strict definition see 2.2). Then a direct connection between
topology and first-order logic is established without the need to go through
the cylindric algebra as an intermediary.

As the title of this paper implies, this paper consists of two parts, one on
the topological representation of the semantics of first-order logic, which
forms the main body of the paper, and the other on the application of this
topological representation as a method in model theory.

As mentioned earlier, the semantic models of propositional logic can be
represented one-to-one as points in Stone spaces, and we want to do something
similar for first-order logic. That is, we want to find a kind of abstractly
defined topological structure such that the construction of such topological
structures can completely and systematically represent the classical semantics
of first-order logic. This work constitutes the first part of this paper,
which consists of Sections 2, 3, and 4.

It is easy to see that the equivalence relation $\sim_{\kappa}$ on the Stone
dual space of a Lindenbaum algebra is transformed into the following
equivalence relation when we turn the Stone dual space into a model space:

$p\sim_{\kappa}q\Leftrightarrow\\{\exists v_{\kappa}\phi:\phi\in p\\}\subseteq
q.$

In fact, we can define a similar equivalence relation $\sim^{\prime}_{\kappa}$
on the cylindric set algebra:

$s\sim^{\prime}_{\kappa}t\Leftrightarrow s\backslash s(\kappa)=t\backslash
t(\kappa).$

The above discussion implies that we can transform the models and theories of
first-order logic into the same class of structures, which should be
topological spaces with a family of additional equivalence relations. In
Section 2.1, we will define these structures strictly and call them cylindric
spaces. In addition, we will define several mappings between cylindric spaces
(Strongly continuous mapping, cylindric mapping, basis-preserving mapping) and
discuss some basic properties of these spaces and mappings. In Section 2.2, as
examples of the cylindric space, we will strictly define the model space and
verify that the cylindric set algebra constructed from a first-order structure
naturally constitutes a cylindric space (called topologization space).
Further, we will also show how to represent the relations between theories and
their models as mappings between spaces.

For a complete topological representation of the semantics of first-order
logic, two issues must be addressed:

  1. 1.

For a theory with infinite models, all of its models constitute a proper
class, so we cannot represent all of them as points in the same space as
propositional logic does.

  2. 2.

Unlike the case of propositional logic, there are formalized connections
between models of first-order logic theories like elementary embeddings, and a
complete topological representation should also consider such connections.

In Section 3, we develop some techniques based on the cylindric space to solve
these problems. The idea is that, for the first problem, there may not be
enough points “in" a space $\mathcal{C}$, But we can construct for each
ordinal $\alpha$ an unique space that is obtained by expanding $\mathcal{C}$
($\alpha$-expansion space of $\mathcal{C}$) so that we have enough points
“associated with" $\mathcal{C}$. For the second problem, we will define a kind
of set-topological relation between points in a cylindric space (factor,
permutation on point) to be used in the next work to represent the elementary
embedding.

In Section 4, based on the previous work, we finally give a topological
representation of the first-order logical semantics. Technically, we build
this representation using the topologization spaces of the first-order
structures mentioned above as mediators. Intuitively, for a theory $T$, all
$T$-models are precisely all first-order structures whose topologization
spaces can be mapped onto the model space of $T$ . Calling the sequences in
the topologization space that list all elements of the first-order structure
as domain points, the points in the model space (or its expansion space) to
which the domain points are mapped can be regarded as representing the first-
order structures. These points representing first-order structures naturally
have a set-topological definition (that is, the set-topological correspondence
property of Henkin set). Based on this, the elementary embeddings between
first-order structures can then be represented as mappings between the
topologization spaces of these first-order structures and thus as relations
between points of the kind we defined in Section 3.

The first part can be seen as part of an overall work: constructing a first-
order analog of the Stone duality. In a forthcoming paper, we will prove that
there is a categorical duality between the category of the cylindric algebra
and the category consisting of the Stone cylindric space and the cylindric
mapping, which will then naturally carry the meaning of the Syntax-Semantics
dual of the first-order logic, based on the work of this paper.

In the second part, Section 5, we introduce a topological method to model
theory systematically based on the first part’s work. In this part, we first
discuss the connection between the model space and the type space, the
classical set-topology concept in model theory. And then, take one theorem in
model theory (which can be regarded as a critical intermediate theorem in the
classical proof of Morley’s theorem) as a example to show the application of
the topological method in model theory. Moreover, we also compare the
topological proof with the corresponding classical proof. As we will see, this
new method greatly reduces the complexity of the proof.

In fact, the features and advantages shown in the example are not limited to
the example. We have already had a complete topological proof of Morley’s
theorem, and the advantages presented in the example are general in this
complete proof (although it is much more compact than the classical proof, it
is still too long to attach here).

It is worth noting that the second part of this paper does not rely very much
on the discussion of the abstract structure of the cylindric space in the
first part. What I mean is that when considering the application of
topological methods to model theory, the only cylindric spaces that need to be
considered are concrete model spaces. Moreover, most of the tools and
properties we introduce in the first part and will use in the application
examples can be constructed or proved on these concrete model spaces by non-
topological and sometimes relatively less complex processes. However, to
simplify the discussion, we will not discuss these non-topological
construction processes in this paper.

##  2 Cylindric space

###  2.1 Basic concept

This section will define some fundamental concepts, including cylindric space,
and discuss their properties. As mentioned in Section 1, cylindric space is a
topological space with additional components. Since the direct definition of
this structure will be very complex, and its property partly depends on the
properties of its components, we first define and discuss these components,
then define the whole structure.

###### Definition 2.1

Let $(S,\tau)$ be a topological space, $\\{\sim_{i}:i\in I\\}$ be an infinite
family of equivalence relations on $S$,

  * •

If for any $a,b\in S$, any $i,j\in I$, $a\sim_{i}c\sim_{j}b$ for some
$c\Leftrightarrow a\sim_{j}c^{\prime}\sim_{i}b$ for some $c^{\prime}$, we say
that $\\{\sim_{i}:i\in I\\}$ is commutative;

  * •

For a basis $B$ of $\tau$, we call $B$ a cylindric basis about
$\\{\sim_{i}:i\in I\\}$ if $B$ is closed under taking union, intersection,
complement and $\sim_{i}$-saturation111For a subset $T\subseteq S$,
$\sim_{i}$-saturation of $T$ writen as $[T]_{i}$ is the set $\\{a\in S:\exists
b\in T,a\sim_{i}b\\}$; If $T=[T]_{i}$, we say $T$ is $\sim_{i}$-saturated..

  * •

We call $\\{\sim_{i}:i\in I\\}$ a cylindric system if it is commutative and
has a cylindric basis.

###### Proposition 2.1

Let $(S,\tau)$ be a topological space, $\\{\sim_{i}:i\in I\\}$ be a cylindric
system on it, we have:

  1. 1.

$(S,\tau)$ is a zero-dimensional space.

  2. 2.

If $(S,\tau)$ is compact, then $\\{u\in\tau:u\mbox{ is a clopen set}\\}$ is
the only cylindric basis about $\\{\sim_{i}:i\in I\\}$.

Proof

  1. 1.

Since the cylindric basis is closed under taking the complement, the first
item holds.

  2. 2.

By definition of the cylindric system, $\\{\sim_{i}:i\in I\\}$ has a cylindric
basis. Choosing an arbitrary cylindric basis $B$ about it, by definition of
cylindric basis, $B\subseteq\\{u\in\tau:u\mbox{ is clopen}\\}$. Conversely,
for an arbitrary clopen set $u\in\tau$, by compactness, there is finite
$B^{\prime}\subseteq B$ s.t. $u=\bigcup B^{\prime}$. By definition of
cylindric basis, $u\in B$.

$\Box$

###### Proposition 2.2

For a topological space $(S,\tau)$, a cylindric system $\\{\sim_{i}:i\in I\\}$
on it and any $U\subseteq\tau$, $i\in I$, we have: $[\bigcup
U]_{i}=\bigcup\\{[u]_{i}:u\in U\\}$.

Proof $a\in[\bigcup U]_{i}\Leftrightarrow\exists b\in\bigcup
U,a\sim_{i}b\Leftrightarrow\exists u\in U,b\in u,a\sim_{i}b\Leftrightarrow
a\in\bigcup\\{[u]_{i}:u\in U\\}$. $\Box$

It is easy to see that the cylindric system is the topological counterpart of
quantifiers. Now we discuss the topological counterpart of equality.

###### Definition 2.2

Let $(S,\tau)$ be a topological space, $\mathcal{E}=\\{\sim_{i}:i\in I\\}$ be
a cylindric system on it, a diagonal family on $\mathcal{E}$ is a family of
clopen sets $\\{D_{ij}:i,j\in I\\}$ satisfying the following conditions: for
any $i,j,k\in I$,

  * •

If $k\in I\backslash\\{i,j\\}$, then $D_{ij}$ is $\sim_{k}$-saturated;

  * •

If $i\neq j$, then for any $a\in S$, $\|[a]_{i}\cap D_{ij}\|=\|[a]_{j}\cap
D_{ij}\|=1$222For a set $X$, by $\|X\|$ we denote the cardinality of $X$; this
implies $S=[D_{ij}]_{i}=[D_{ij}]_{j}$;

  * •

$D_{ij}\cap D_{jk}\subseteq D_{ik}$, $D_{ij}=D_{ji}$.

We call elements in the diagonal family diagonals.

###### Lemma 2.1

Let $\\{D_{ij}:i,j\in I\\}$ be a diagonal family on some cylindric system on
some topological space, for any $i,j,k\in I$,

  1. 1.

For any $u$, if $k\not\in\\{i,j\\}$, then $[D_{ij}\cap
u]_{k}=D_{ij}\cap[u]_{k}$;

  2. 2.

$D_{ii}=S$; (and therefore, if $i=j$ or $k=j$, then $D_{ik}=D_{ij}\cap
D_{jk}$)

  3. 3.

If $j\not\in\\{i,k\\}$, then $D_{ik}=[D_{ij}\cap D_{jk}]_{j}$.

Proof

  1. 1.

     * •

$\subseteq$: If $a\in[D_{ij}\cap u]_{k}$, we know that there is $b\sim_{k}a$
s.t. $b\in D_{ij}\cap u$, by definition of $D_{ij}$, $a\in D_{ij}$, then $a\in
D_{ij}\cap[D_{ij}\cap u]_{k}\subseteq D_{ij}\cap[u]_{k}$;

     * •

$\supseteq$: If $a\in D_{ij}\cap[u]_{k}$, then $a\in D_{ij}$ and there is
$b\sim_{k}a$ s.t. $b\in u$, by definition of $D_{ij}$, $b\in D_{ij}$, then
$a\in[D_{ij}\cap u]_{k}$.

  2. 2.

Choose an arbitrary $i^{\prime}\neq i$, we have $D_{ii}$ is
$\sim_{i^{\prime}}$-saturated, then
$S=[D_{ii^{\prime}}]_{i^{\prime}}=[D_{ii^{\prime}}\cap
D_{i^{\prime}i}]_{i^{\prime}}\subseteq[D_{ii}]_{i^{\prime}}=D_{ii}$. Hence
$D_{ii}=S$.

  3. 3.

     * •

Case 1: $i\neq k$.

Clearly we have $[D_{ij}\cap D_{jk}]_{j}\subseteq[D_{ik}]_{j}=D_{ik}$.

For any $a\in D_{ij}$, $[a]_{k}\subseteq D_{ij}$, then there is $b\sim_{k}a$
s.t. $b\in D_{ij}\cap D_{jk}$. Hence $D_{ij}\subseteq[D_{ij}\cap D_{jk}]_{k}$.
By commutativity of cylindric system, $S=[D_{ij}]_{j}\subseteq[[D_{ij}\cap
D_{jk}]_{k}]_{j}=[[D_{ij}\cap D_{jk}]_{j}]_{k}$ i.e. $[[D_{ij}\cap
D_{jk}]_{j}]_{k}=S$. This means that for any $a,[D_{ij}\cap
D_{jk}]_{j}\cap[a]_{k}\neq\emptyset$, then $[D_{ij}\cap
D_{jk}]_{j}\cap[a]_{k}=D_{ik}\cap[a]_{k}$. Hence $[D_{ij}\cap
D_{jk}]_{j}=D_{ik}$.

     * •

Case 2: $i=k$.

$[D_{ij}\cap D_{jk}]_{j}=[D_{ij}]_{j}=S=D_{ii}=D_{ik}$.

$\Box$

Now, let’s introduce the primary object that the subsequent work will focus
on: cylindric space.

###### Definition 2.3

A cylindric space is a triad $(\mathcal{S},\mathcal{E}_{\alpha},\mathcal{D})$,
where $\mathcal{S}$ is a topological space, $\mathcal{E}_{\alpha}$ is a
cylindric system on $\mathcal{S}$ with infinite ordinal $\alpha$ as index,
$\mathcal{D}$ is a diagonal family on $\mathcal{E}_{\alpha}$.

For a cylindric space $\mathcal{C}$, we write its underlying set as
$|\mathcal{C}|$, underlying topology as $\tau^{\mathcal{C}}$, cylindric system
as $\mathcal{E}^{\mathcal{C}}$ and equivalence relations in it as
$\sim^{\mathcal{C}}_{i}$, diagonal family as $\mathcal{D}^{\mathcal{C}}$ and
diagonals in it as $D^{\mathcal{C}}_{ij}$.

###### Definition 2.4

  * •

For an arbitrary cylindric space $\mathcal{C}$, we call the index ordinal
$\alpha$ of $\mathcal{E}^{\mathcal{C}}$ the dimension of $\mathcal{C}$. For
convenience, we sometimes write $\mathcal{C}$ as $\mathcal{C}_{\alpha}$ to
indicate that its dimension is $\alpha$;

  * •

For any arbitrary subset $X\subseteq|\mathcal{C}|$, define
$\Delta(X):=\\{i\in\alpha:X$ is not $\sim_{i}$-saturated$\\}$, we call
$\Delta(X)$ the dimension of $X$;

  * •

If there is a cylindric basis $B$ of $\mathcal{C}$ s.t. for any $u\in B$,
$\|\Delta(u)\|<\omega$, then we say $\mathcal{C}$ is basis-finite.

Now, we define the mapping between cylindric spaces.

###### Definition 2.5

  * •

Let $\alpha\geq\beta$ be two ordinals,
$\mathcal{C}_{\alpha},\mathcal{C^{\prime}}_{\beta}$ be two cylindric space,
$f:|\mathcal{C}|\rightarrow|\mathcal{C^{\prime}}|$ be a mapping between them,
we call $f$ a strongly continuous mapping (S-mapping for short) and denote it
as $f:\mathcal{C}\rightarrow\mathcal{C^{\prime}}$ if the following two
conditions are satisfied:

    * –

$f:(|\mathcal{C}|,\tau^{\mathcal{C}})\rightarrow(|\mathcal{C^{\prime}}|,\tau^{\mathcal{C^{\prime}}})$
is continuous where
$f^{-1}[D^{\mathcal{C^{\prime}}}_{ij}]=D^{\mathcal{C}}_{ij}$ for any
$i,j\in\beta$;

    * –

$f:(|\mathcal{C}|,\sim_{i})_{i\in\beta}\rightarrow(|\mathcal{C^{\prime}}|,\sim_{i})_{i\in\beta}$
is a structure homomorphism;

  * •

For an S-mapping $f$, if $f^{-1}$ is also an S-mapping, we call $f$ an
S-homeomorphism, and denote it as
$f:\mathcal{C}_{\alpha}\cong\mathcal{C^{\prime}}_{\beta}$;

  * •

We call an S-mapping
$f:\mathcal{C}_{\alpha}\rightarrow\mathcal{C^{\prime}}_{\beta}$ a cylindric
mapping (C-mapping for short), if for any $i\in\alpha$, any
$a\in|\mathcal{C}|$, $f[[a]_{i}]$ is a dense subset of $[f(a)]_{i}$.

  * •

Let $f:\mathcal{C}_{\alpha}\rightarrow\mathcal{C^{\prime}}_{\beta}$ be an
S-mapping, we say $f$ is basis-preserving, if for any $u\in\tau^{\mathcal{C}}$
with $\Delta(u)\subseteq\beta$, there is a
$u^{\prime}\in\tau^{\mathcal{C^{\prime}}}$ s.t. $f^{-1}[u^{\prime}]=u$.

###### Proposition 2.3

Let $f:\mathcal{C}_{\alpha}\rightarrow\mathcal{C^{\prime}}_{\beta}$ be an
S-mapping.

  1. 1.

For any $u^{\prime}\in\tau^{\mathcal{C^{\prime}}}$,
$\Delta(f^{-1}[u^{\prime}])\subseteq\Delta(u^{\prime})$;

  2. 2.

The following statements are equivalent:

     * •

$f$ is a C-mapping.

     * •

For any $u^{\prime}\in\tau^{\mathcal{C^{\prime}}}$, $i\in\alpha$,
$[f^{-1}[u^{\prime}]]_{i}=f^{-1}[[u^{\prime}]_{i}]$.

     * •

For any clopen set $u^{\prime}$ in a cylindric basis, $i\in\alpha$,
$[f^{-1}[u^{\prime}]]_{i}=f^{-1}[[u^{\prime}]_{i}]$.

Proof

  1. 1.

For any $i\in\beta$,

| $i\not\in\Delta(u^{\prime})$
---|---
$\Leftrightarrow$ | for any $a^{\prime}\in u^{\prime},b^{\prime}\in-u^{\prime},a\not\sim_{i}b$
$\Rightarrow$ | for any $a\in f^{-1}[u^{\prime}],b^{\prime}\in f^{-1}[-u^{\prime}]=-f^{-1}[u^{\prime}],a\not\sim_{i}b$
$\Leftrightarrow$ | $i\not\in\Delta(f^{-1}[u^{\prime}])$

For any $i\in\alpha\backslash\beta$, clearly
$i\not\in\Delta(f^{-1}[u^{\prime}])$ since the image of any
$\sim_{i}$-equivalence class is a singleton. Hence,
$\Delta(f^{-1}[u^{\prime}])\subseteq\Delta(u^{\prime})$.

  2. 2.

Since for any collection $U$ of open sets, $[\bigcup
U]_{i}=\bigcup\\{[u]_{i}:u\in U\\}$, the second equivalence holds. By
Proposition 2.3.(1), if $i\in\alpha\backslash\beta$, the first equivalence
obviously holds. If $i\in\beta$, it is also easy to see
$[f^{-1}[u^{\prime}]]_{i}\subseteq f^{-1}[[u^{\prime}]_{i}]$ holds for any
S-mapping $f$. Then we have

|  $f$ is not a C-mapping
---|---
$\Leftrightarrow$ | there exists $i\in\alpha$, $u^{\prime}\in\tau^{\mathcal{C^{\prime}}}$ and $a\in|\mathcal{C}|$, $a\in f^{-1}[[u^{\prime}]_{i}]$ but $[a]_{i}\cap f^{-1}[u^{\prime}]=\emptyset$
$\Leftrightarrow$ | there exists $i\in\alpha$, $u^{\prime}\in\tau^{\mathcal{C^{\prime}}}$ and $a\in|\mathcal{C}|$, $a\in f^{-1}[[u^{\prime}]_{i}]$ but $a\not\in[f^{-1}[u^{\prime}]]_{i}$
$\Leftrightarrow$ | there exists $i\in\alpha$, $u^{\prime}\in\tau^{\mathcal{C^{\prime}}}$, $[f^{-1}[u^{\prime}]]_{i}\neq f^{-1}[[u^{\prime}]_{i}]$

$\Box$

###### Proposition 2.4

Let
$f:\mathcal{C}_{\alpha}\rightarrow\mathcal{C^{\prime}}_{\alpha^{\prime}},g:\mathcal{C^{\prime}}_{\alpha^{\prime}}\rightarrow\mathcal{C^{\prime\prime}}_{\alpha^{\prime\prime}}$
be two S-mapping, then $g\circ f$ is an S-mapping. This statement also holds
for C-mapping and basis-preserving C-mapping.

Proof Clearly $g\circ f$ is continuous and for any
$i,j\in\alpha^{\prime\prime}$, $(g\circ
f)^{-1}[D^{\mathcal{C^{\prime\prime}}}_{ij}]=D^{\mathcal{C}}_{ij}$. By the
property of structural homomorphism, $g\circ f$ also forms structural
homomorphism for corresponding equivalence relations. Hence, $g\circ f$ is an
S-mapping.

  * •

C-mapping: For any $u\in\tau^{\mathcal{C^{\prime\prime}}}$, $i\in\alpha$, By
Proposition 2.3.(2), we have $(g\circ
f)^{-1}[[u]_{i}]=f^{-1}[g^{-1}[[u]_{i}]]=f^{-1}[[g^{-1}[u]]_{i}]=[f^{-1}[g^{-1}[u]]]_{i}=[(g\circ
f)^{-1}[u]]_{i}$ and then $g\circ f$ is a C-mapping. Hence, the case about
C-mapping holds.

  * •

basis-preserving C-mapping: For any $u\in\tau^{\mathcal{C}}$ with
$\Delta(u)\subseteq\alpha^{\prime\prime}$, there is
$u^{\prime}\in\tau^{\mathcal{C^{\prime}}}$ s.t. $f^{-1}[u^{\prime}]=u$. For
any finite
$s=\\{s_{1},...,s_{n}\\}\subseteq\Delta(u^{\prime})\backslash\Delta(u)$,
abbreviating $[...[u^{\prime}]_{s_{1}}...]_{s_{n}}$ to $[u^{\prime}]_{s}$, by
Proposition 2.3 (2), we know
$f^{-1}[[u^{\prime}]_{s}]=[f^{-1}[u^{\prime}]]_{s}=[u]_{s}=u$. Let
$X:=\Delta(u^{\prime})\backslash\alpha^{\prime\prime}$,
$v^{\prime}:=\bigcup\\{[u^{\prime}]_{s}:s\subseteq X$ is finite$\\}$, we know
$v^{\prime}$ is open, $\Delta(v^{\prime})\subseteq\alpha^{\prime\prime}$ and
$f^{-1}[v^{\prime}]=u$. Then there is
$u^{\prime\prime}\in\tau^{\mathcal{C^{\prime\prime}}}$ s.t.
$g^{-1}[u^{\prime\prime}]=u^{\prime}$ then $(g\circ
f)^{-1}[u^{\prime\prime}]=u$. Hence, the case about basis-preserving C-mapping
holds.

$\Box$

###### Lemma 2.2

Basis-preserving S-bijection between two cylindric spaces with the same
dimension is S-homeomorphism.

Proof Let $f:\mathcal{C}_{\alpha}\rightarrow\mathcal{C^{\prime}}_{\alpha}$ be
a basis-preserving S-bijection, We show $f^{-1}$ is also a S-mapping:

It’s easy to see that
$(f^{-1})^{-1}[D^{\mathcal{C}}_{ij}]=D^{\mathcal{C^{\prime}}}_{ij}$ and since
$f$ is basis-preserving, $f^{-1}$ is continuous.

For any $a^{\prime},b^{\prime}\in|\mathcal{C^{\prime}}|$, any $i\in\alpha$, if
$a^{\prime}\sim_{i}b^{\prime}$, then there is an unique
$c^{\prime}\sim_{i}a^{\prime},b^{\prime}$ s.t. $c^{\prime}\in
D^{\mathcal{C^{\prime}}}_{ij}$. Assume
$f^{-1}(a^{\prime})\not\sim_{i}f^{-1}(b^{\prime})$, then there are different
$c\sim_{i}f^{-1}(a^{\prime}),d\sim_{i}f^{-1}(b^{\prime})$ s.t. $c,d\in
D^{\mathcal{C}}_{ij}$. By definition of S-mapping,
$f(c)\in[a^{\prime}]_{i}\cap
D^{\mathcal{C^{\prime}}}_{ij}=[b^{\prime}]_{i}\cap
D^{\mathcal{C^{\prime}}}_{ij}\ni f(d)$, which means $f(c)=c^{\prime}=f(d)$.
This contradicts with bijectivity and therefore
$f^{-1}(a^{\prime})\sim_{i}f^{-1}(b^{\prime})$. $\Box$

###### Proposition 2.5

Let $\mathcal{C}_{\alpha}$ be a $T_{2}$ cylindric space, $i\in\alpha$, then
$[a]_{i}$ is a closed set for any $a\in|\mathcal{C}|$ and if $\mathcal{C}$ is
compact, $[u]_{i}$ is closed for any closed set $u\subseteq|\mathcal{C}|$.

Proof Let $U$ be the family of all clopen neighborhoods of $[u]_{i}$. We know
that $[u]_{i}\subseteq\bigcap U$. For any $v\in U$, we know that
$-v\cap[u]_{i}=\emptyset$, then $[-v]_{i}\cap[u]_{i}=\emptyset$, then
$[u]_{i}\subseteq-([-v]_{i})\subseteq v$ and $-([-v]_{i})\subseteq U$ is
$\sim_{i}$-saturated. Write $U^{\prime}:=\\{v\in U:v$ is
$\sim_{i}$-saturated$\\}$, we know $\bigcap U=\bigcap U^{\prime}$. So we just
need to show $[u]_{i}\not\subset\bigcap U^{\prime}$ to prove $[u]_{i}$ is
closed:

  * •

Firstly, consider the case that $u$ is a singleton. Write $u=\\{a\\}$:

Assume there exists $b\in\bigcap U^{\prime}\backslash[u]_{i}$, we know
$[b]_{i}\subseteq\bigcap U^{\prime}$ and $[b]_{i}\cap[u]_{i}=\emptyset$.
Choose an arbitrary $j\neq i$, we know
$\|D_{ij}\cap[a]_{i}\|=\|D_{ij}\cap[b]_{i}\|=1$. Let $a^{\prime}\in
D_{ij}\cap[a]_{i},b^{\prime}\in D_{ij}\cap[b]_{i}$.

For any clopen $v$, if $a^{\prime}\in v$, then $a\in[v\cap D_{ij}]_{i}$. Since
$[v\cap D_{ij}]_{i}\in U^{\prime}$, we know $b^{\prime}\in[v\cap D_{ij}]_{i}$,
then $b^{\prime}\in[v\cap D_{ij}]_{i}\cap D_{ij}$ i.e., $\exists
b^{\prime\prime}\sim_{i}b^{\prime}$, $b^{\prime\prime}\in v\cap D_{ij}$ and
$b^{\prime}\in D_{ij}$. By the fact $\|D_{ij}\cap[b^{\prime}]_{i}\|=1$, we
know $b^{\prime}=b^{\prime\prime}$, then $b^{\prime}\in v$, vice versa. Since
we choose $v$ arbitrarily, $\\{w\in\tau:w\mbox{ is clopen and }a^{\prime}\in
w\\}=\\{w\in\tau:w\mbox{ is clopen and }b^{\prime}\in w\\}$. By property of
$T_{2}$ space, $a^{\prime}=b^{\prime}$ which contradicts the assumption. Hence
there isn’t $b\in\bigcap U^{\prime}\backslash[u]_{i}$, then
$[u]_{i}\not\subset\bigcap U^{\prime}$.

  * •

Now, consider the general $u$:

Assume there exists $b\in\bigcap U^{\prime}\backslash[u]_{i}$, we know
$[b]_{i}\subseteq\bigcap U^{\prime}$ and $[b]_{i}\cap[u]_{i}=\emptyset$. For
any clopen $v\supseteq u$, we know $[v]_{i}\in U^{\prime}$ and therefore
$[b]_{i}\subseteq[v]_{i}$, then $v\cap[b]_{i}\neq\emptyset$. Since
$\\{w\in\tau^{\mathcal{C}}:w\mbox{ is clopen and }u\subseteq w\\}$ is closed
under taking intersect and $[b]_{i}$ is closed set, by compactness we know
$\bigcap\\{w\in\tau^{\mathcal{C}}:w\mbox{ is clopen and }u\subseteq
w\\}\cap[b]_{i}=u\cap[b]_{i}\neq\emptyset$, then $b\in[u]_{i}$ which
contradicts assumption. Hence $[u]_{i}\not\subset\bigcap U^{\prime}$.

$\Box$

###  2.2 Four examples

In the previous subsection, we have defined cylindric spaces and the mappings
between them. As introduced in the Introduction, we are trying to describe the
structure of first-order logical semantics in the topological language through
cylindric space. In this subsection, we respectively use the first-order
structure and formulas to construct two specific examples of cylindric space
and two specific examples of S-mapping.

#### Example 1:

Let $\mathcal{L}$ be a first-order language, $\Lambda_{\mathcal{L}}$ be the
set of $\mathcal{L}$-formulas, $\\{R_{i}:i\in I\\}$ be all predicates in
$\mathcal{L}$ (as mentioned in Introduction, the only non-logical symbols in
$\mathcal{L}$ are predicates), $\mathfrak{A}=(A,R^{\mathfrak{A}}_{i})_{i\in
I}$ be an $\mathcal{L}$-structure.

Let $A^{\omega}$ be the $\omega$ power of $A$, for any formula
$\phi\in\Lambda_{\mathcal{L}}$, we define $A_{\phi}\subseteq A^{\omega}$ as:
$A_{\phi}=\\{a:$ for every free variables $x_{i_{1}},...,x_{i_{n}}\mbox{ in
}\phi,\langle
a({i_{1}})...a({i_{n}})\rangle\in\phi(\mathfrak{A})\\}$333$\phi(\mathfrak{A})$
denotes the subset defined by $\phi$ on $\mathfrak{A}$. Let
$B:=\\{A_{\phi}:\phi\in\Lambda_{\mathcal{L}}\\}$, it is easy to see that both
empty set, and universal set are in $B$ and $B$ is closed under taking
intersection, union and complement, empty set, and universal set. Hence $B$
forms a basis of a zero-dimensional topology on $A^{\omega}$. We denote this
topology as $\tau^{A}$.

For each $i\in\omega$, we define equivalence relation
$\sim^{\mathfrak{A}}_{i}$ on $A^{\omega}$ as:
$a\sim^{\mathfrak{A}}_{i}b\Leftrightarrow$ for any $j\neq i$, $a(j)=b(j)$. Let
$\mathcal{S^{\mathfrak{A}}}:=(A^{\omega},\tau^{A})$, for any $i,j\in\omega$,
$D^{\mathfrak{A}}_{ij}:=A_{v_{i}=v_{j}}$,
$\mathcal{C^{\mathfrak{A}}_{\omega}}:=(\mathcal{S^{\mathfrak{A}}},\\{\sim^{\mathfrak{A}}_{i}:i\in\omega\\},\\{D^{\mathfrak{A}}_{ij}:i,j\in\omega\\})$.
We have:

###### Proposition 2.6

$\mathcal{C^{\mathfrak{A}}_{\omega}}$ is a basis-finite cylindric space.

Proof

  * •

$\\{\sim^{A}_{i}:i\in\omega\\}$ is a cylindric system:

$[[a]_{i}]_{j}=\\{b:$ for $k\not\in\\{i,j\\},a(k)=b(k)\\}=[[a]_{j}]_{i}$, so
$\\{\sim^{\mathfrak{A}}_{i}:i\in\omega\\}$ is commutative. For any $u,v\in B$,
we know there exist formulas $\phi,\psi$ s.t. $u=A_{\phi},v=A_{\psi}$. It is
easy to see $-A_{\phi}=A_{\neg\phi},A_{\phi}\cap
A_{\psi}=A_{\phi\wedge\psi},A_{\phi}\cup A_{\psi}=A_{\phi\vee\psi}$ and for
any $i\in\omega$, $[A_{\phi}]_{i}=A_{\exists v_{i}\phi}$. Hence $B$ is closed
under taking intersect, union, complement and
$\sim^{\mathfrak{A}}_{i}$-saturation. Hence
$\\{\sim^{\mathfrak{A}}_{i}:i\in\omega\\}$ forms a cylindric system.

  * •

$\\{D^{\mathfrak{A}}_{ij}:i,j\in\omega\\}$ is a diagonal family on
$\\{\sim^{A}_{i}:i\in\omega\\}$:

For any pair $i\neq j\in\omega$, any $a\in A^{\omega}$, we define $a_{ij}$ as:
$a_{ij}(i)=a(j)$ and for any $k\neq i$, $a_{ij}(k)=a(k)$. By definition of
$D^{\mathfrak{A}}_{ij}$, we know $a_{ij}$ is the only point in
$D^{\mathfrak{A}}_{ij}\cap[a]_{i}=D^{\mathfrak{A}}_{ji}\cap[a]_{i}$. Hence the
intersection of $D^{\mathfrak{A}}_{ij}=D^{\mathfrak{A}}_{ji}$ and any
$\sim^{\mathfrak{A}}_{i}$-equivalence class is singleton. For any
$k\not\in\\{i,j\\}$ and any $a^{\prime}\in[a]_{k}$, we know
$a^{\prime}(i)=a(i)=a(j)=a^{\prime}(j)$ and therefore $a^{\prime}\in
D^{\mathfrak{A}}_{ij}$. Hence $[a]_{k}\subseteq D^{\mathfrak{A}}_{ij}$. Hence
$D^{\mathfrak{A}}_{ij}$ is $\sim_{k}$-saturated.

For any $i,j,k\in\omega$, $a\in D^{\mathfrak{A}}_{ij}\cap
D^{\mathfrak{A}}_{jk}\Rightarrow a(i)=a(j)=a(k)\Rightarrow a\in
D^{\mathfrak{A}}_{ik}$. Hence $D^{\mathfrak{A}}_{ij}\cap
D^{\mathfrak{A}}_{jk}\subseteq D^{\mathfrak{A}}_{ik}$.

  * •

For any $u\in B$, $|\Delta(u)|<\omega$:

For any $u\in B$, we know there is a formula $\phi$ s.t. $u=A_{\phi}$. For any
$a\in u$ and variable $v_{i}$ that doesn’t occur freely in $\phi$, it’s easy
to see that $[a]_{i}\subseteq A_{\phi}=u$ i.e. $u$ is
$\sim^{\mathfrak{A}}_{i}$-saturated. Since there are only finitely many free
variables in $\phi$, we know $\Delta(u)<\omega$.

$\Box$

We say that $\mathcal{C^{\mathfrak{A}}_{\omega}}$ is the
$\omega$-topologization of $\mathfrak{A}$. We call a sequence $a\in
A^{\omega}$ which lists all the elements of $\mathfrak{A}$ a domain point of
$\mathcal{C^{\mathfrak{A}}_{\omega}}$. For an assignment $\sigma$ and a
sequence $a\in A^{\omega}$, if for each $i\in\omega$, $a(i)=\sigma(v_{i})$,
then we say that $a$ represent $\sigma$.

#### Example 2:

For any langage $\mathcal{L}$, any $\mathcal{L}$-theory $T$ and any
$\Phi\subseteq\Lambda_{\mathcal{L}}$, if there is $T$-model $\mathfrak{A}$ and
assignment $\sigma$ on $\mathfrak{A}$ s.t.
$\Phi=\\{\phi\in\Lambda_{\mathcal{L}}:(\mathfrak{A},\sigma)\vDash\phi\\}$, we
call $\Phi$ a maximal satisfiable set (MSS) of $T$.

Let $S_{T}$ be the family of all MSSs of $T$. For each formula $\phi$, we
define $S_{\phi}:=\\{a\in S_{T}:\phi\in a\\}$. It is easy to see that
$B=\\{S_{\phi}:\phi\in\Lambda_{\mathcal{L}}\\}$ is a topological basis, we
write the induced topology as $\tau^{T}$.

For each $i\in\omega$, we define equivalence relation $\sim^{T}_{i}$ as:
$a\sim_{i}b\Leftrightarrow\\{\phi\in a:v_{i}$ doesn’t occur in $\phi$
freely$\\}=\\{\phi\in b:v_{i}$ doesn’t occur in $\phi$ freely$\\}$. For each
$i,j\in\omega$, we define $D^{T}_{ij}:=S_{v_{i}=v_{j}}$ and write
$\mathcal{C}^{T}=((S_{T},\tau^{T}),\\{\sim^{T}_{i}:i\in\omega\\},\\{D^{T}_{ij}:i,j\in\omega\\})$.
We have:

###### Proposition 2.7

$\mathcal{C}^{T}$ is a $T_{2}$ basis-finite cylindric space.

Proof

  * •

$\\{\sim^{T}_{i}:i\in\omega\\}$ is a cylindric system:

For any $a,b\in S_{T}$,

| $b\in[[a]_{i}]_{j}$
---|---
$\Leftrightarrow$ | for any formula $\phi$ satisfying $v_{i},v_{j}$ doesn’t occur in it freely, we have $\phi\in a$ if and only if $\phi\in b$
$\Leftrightarrow$ | $b\in[[a]_{j}]_{i}$

Hence $\\{\sim^{T}_{i}:i\in\omega\\}$ is commutative.

For any $u,v\in B$, there are formulas $\phi,\psi$ s.t.
$u=S_{\phi},v=S_{\psi}$. It is easy to see
$-S_{\phi}=S_{\neg\phi},S_{\phi}\cap S_{\psi}=S_{\phi\wedge\psi},S_{\phi}\cup
S_{\psi}=S_{\phi\vee\psi}$ and for any $i\in\omega$,
$[S_{\phi}]_{i}=S_{\exists v_{i}\phi}$. Hence $B$ is closed under
intersection, union, complement and $\sim^{T}_{i}$-saturation.

Hence, $\\{\sim^{T}_{i}:i\in\omega\\}$ is a cylindric system.

  * •

$\\{D^{T}_{ij}:i,j\in\omega\\}$ is a diagonal family:

For any pair $i\neq j\in\omega$ and any $a\in S_{T}$, there is a $T$-model
$\mathfrak{A}$ and an assignment $\sigma$ on $\mathfrak{A}$ s.t.
$(\mathfrak{A},\sigma)\vDash a$, we define assignment $\sigma^{\prime}$ as:
$\sigma^{\prime}(v_{i})=\sigma(v_{j})$ and for any $k\neq i$,
$\sigma^{\prime}(v_{k})=\sigma(v_{k})$. Let
$a_{\sigma^{\prime}}:=\\{\phi:(\mathfrak{A},\sigma^{\prime})\vDash\phi\\}$, we
know $a_{\sigma^{\prime}}\in[a]_{i}$ and is the only point containing
$v_{i}=v_{j}$ in $[a]_{i}$. Hence $\|D^{T}_{ij}\cap[a]_{i}\|=1$. For any
$k\not\in\\{i,j\\}$, any $a\in D^{T}_{ij}$ and any $a^{\prime}\in[a]_{k}$, by
definition, $v_{i}=v_{j}\in a^{\prime}$, then $a^{\prime}\in D^{T}_{ij}$.
Hence $[a]_{k}\subseteq D^{T}_{ij}$. Hence $D^{\mathfrak{A}}_{ij}$ is
$\sim^{T}_{k}$-saturated.

For any $i,j,k\in\omega$, $a\in D^{T}_{ij}\cap
D^{T}_{jk}\Rightarrow\\{v_{i}=v_{j},v_{j}=v_{k}\\}\subseteq
a\Rightarrow\\{v_{i}=v_{k}\\}\subseteq a\Rightarrow a\in D^{T}_{ik}$. Hence
$D^{T}_{ij}\cap D^{T}_{jk}\subseteq D^{T}_{ik}$.

  * •

For any $u\in B$, $\Delta(u)<\omega$:

For any $u\in B$, there is a formula $\phi$ s.t. $u=S_{\phi}$. For any $a\in
u$ and variable $v_{i}$ which does not occur freely in $\phi$, by definition
we have $[a]_{i}\subseteq S_{\phi}=u$ i.e. $u$ is $\sim^{T}_{i}$-saturated.
Since there are only finitely many free variables in $\phi$, we know
$\Delta(u)<\omega$.

  * •

$T_{2}$ property:

For any pair $a\neq b\in S_{T}$, we know there exists a formula $\phi$ s.t.
$\phi\in a,\neg\phi\in b$, then $a\in S_{\phi},b\in S_{\neg\phi}$. Since
$S_{\phi}\cap S_{\neg\phi}=\emptyset$, this means the property of $T_{2}$.

$\Box$

We call $\mathcal{C}^{T}$ the model space of $T$. It is easy to see from the
compactness theorem of first-order logic that any clopen covering in the model
space of $T$ must have a finite sub-covering, which means that the model space
of $T$ is compact and thus is an FOL space (Definition 3.3 below). We are not
in a hurry to write this property into the above proposition here because we
will give a topological proof of this property in Theorem 4.1.

###### Remark 2.1

As mentioned in the Introduction, the work of this paper is inspired by the
structure of cylindric set algebra and Charles Pinter’s research on Stone dual
space of cylindric algebra. This inspiration is embodied in the above two
examples. Among them, Example 1 can be regarded as the topological structure
naturally derived from the cylindric set algebra on $\mathfrak{A}$; the second
example is essentially the “cylindric dual space” in Charles Pinter’s article.

#### Example 3

Let $\mathcal{L},\mathcal{L^{\prime}}$ be two first-order languages s.t.
$\mathcal{L^{\prime}}\leq\mathcal{L}$, $T$ be an $\mathcal{L}$-theory,
$T^{\prime}$ be an $\mathcal{L^{\prime}}$-theory s.t.
$T^{\prime}\subseteq\\{\phi\in\Lambda_{\mathcal{L^{\prime}}}:\phi\in T\\}$.
Let $f:|\mathcal{C}^{T}|\rightarrow|\mathcal{C}^{T^{\prime}}|$ be
$f(a)=a|_{\mathcal{L^{\prime}}}$
($a|_{\mathcal{L^{\prime}}}:=\\{\phi\in\Lambda_{\mathcal{L^{\prime}}}:\phi\in
a\\}$), then we have:

###### Proposition 2.8

$f:\mathcal{C}^{T}\rightarrow\mathcal{C}^{T^{\prime}}$ is a C-mapping and if
$\mathcal{L}=\mathcal{L^{\prime}}$, then $f$ is basis-preserving.

Proof

  * •

continuity and preserving diagonal family:

Here we only consider the case about the clopen sets in the cylindric basis
(i.e., $\\{S^{\prime}_{\phi}:\phi\in\Lambda_{\mathcal{L^{\prime}}}\\}$), the
case about general open sets is an obvious corollary.

For any formula $\phi\in\Lambda_{\mathcal{L^{\prime}}}$,

| $a\in f^{-1}[S^{\prime}_{\phi}]$
---|---
$\Leftrightarrow$ | $\phi\in f(a)$
$\Leftrightarrow$ | $\phi\in a$
$\Leftrightarrow$ | $a\in S_{\phi}$

Hence $f^{-1}[S^{\prime}_{\phi}]=S_{\phi}\in\tau^{\mathcal{C}^{T}}$ and
particularly
$f^{-1}[D^{\mathcal{C}^{T^{\prime}}}_{ij}]=D^{\mathcal{C}^{T}}_{ij}$.

  * •

structure homomorphism:

For any $a,b\in S_{T}$, $i\in\omega$, we have

| $a\sim_{i}b$
---|---
$\Leftrightarrow$ | for any formula $\phi\in\Lambda_{\mathcal{L}}$ in which $v_{i}$ doesn’t occur freely, $\phi\in a$ if and only if $\phi\in b$
$\Rightarrow$ | for any formula $\phi\in\Lambda_{\mathcal{L^{\prime}}}$ in which $v_{i}$ doesn’t occur freely, $\phi\in f(a)$ if and only if $\phi\in f(b)$
$\Leftrightarrow$ | $f(a)\sim_{i}f(b)$

  * •

C-mapping:

For any $a\in|\mathcal{C}|$ and any
$S^{\prime}_{\phi}\cap[f(a)]_{i}\neq\emptyset$, we know
$f(a)\in[S^{\prime}_{\phi}]_{i}=S^{\prime}_{\exists v_{i}\phi}$, then $a\in
S_{\exists v_{i}\phi}=[S_{\phi}]_{i}$, then there is $b\in[a]_{i}\cap
S_{\phi}$, then $f(b)\in S^{\prime}_{\phi}\cap[f(a)]_{i}$. hence, $f[[a]_{i}]$
is dense in $[f(a)]_{i}$.

  * •

If $\mathcal{L}=\mathcal{L^{\prime}}$

Then for any $\phi\in\Lambda_{\mathcal{L}}$,
$S_{\phi}=f^{-1}[S^{\prime}_{\phi}]$, then $f$ is basis-preserving.

$\Box$

#### Example 4

Let $\mathcal{L}$ be a first-order language, $T$ be an $\mathcal{L}$-theory,
$\mathfrak{A}$ be a $T$-model. Define
$f:|\mathcal{C}^{\mathfrak{A}}_{\omega}|\rightarrow\mathcal{|}{C}_{T}|$ as
$f(a)=\\{\phi:(\mathfrak{A},\sigma_{a})\vDash\phi\\}$ ($\sigma_{a}$ is the
assignment represented by $a$), we have:

###### Proposition 2.9

$f:\mathcal{C}^{\mathfrak{A}}_{\omega}\rightarrow\mathcal{C}_{T}$ is basis-
preserving C-mapping.

Proof

  * •

continuity and diagonal-preserving:

Here we only consider the case about the clopen sets in cylindric basis (i.e.
$\\{S_{\phi}:\phi\in\Lambda_{\mathcal{L}}\\}$ and
$\\{A_{\phi}:\phi\in\Lambda_{\mathcal{L}}\\}$), the case about general open
sets is an obvious corollary.

For any $\phi\in\Lambda_{\mathcal{L}}$,

| $a\in f^{-1}[S_{\phi}]$
---|---
$\Leftrightarrow$ | $\phi\in f(a)$
$\Leftrightarrow$ | $(\mathfrak{A},\sigma_{a})\vDash\phi$
$\Leftrightarrow$ | $a\in A_{\phi}$

Hence $f^{-1}[S_{\phi}]=A_{\phi}$ and particularly,
$f^{-1}[D^{T}_{ij}]=D^{\mathfrak{A}}_{ij}$.

  * •

structure homomorphism, C-mapping and basis-preserving: the proof is the same
as the corresponding part of the proof of Example 3, except that $S_{\phi}$
needs to be changed to $A_{\phi}$, and $S^{\prime}_{\phi}$ to $S_{\phi}$.

$\Box$

##  3 Some Basic Works

As mentioned in Introduction, the topological representation of semantics of
first-order logic based on the cylindric space faces two requirements: we need
a point-to-point relation on the space to represent elementary embeddings
between models and we need enough points to represent all models of a theory.

This section will devote to providing the basic topological tools to meet
these requirements.

###  3.1 Permutation on sets and points

In this subsection, we introduce a kind of point-to-point, set-to-set
relations on the basis-finite cylindric space, namely permutation and factor.
Intuitively, this kind of relationship can be regarded as an abstract
characterization of the endo-functions on the domain of first-order structure.

For convenience, we agree on the following notation:

###### Definition 3.1

For any cylindric space $\mathcal{C}_{\alpha}$, $u\subseteq|\mathcal{C}|$ and
$i,j\in\alpha$, if $i\neq j$, we write $u(\frac{i}{j}):=[u\cap
D^{\mathcal{C}}_{ij}]_{j}$, if $i=j$, we write $u(\frac{i}{j}):=u$.

Now, let us do some technical preparations:

###### Proposition 3.1

Let $\mathcal{C}_{\alpha}$ be a cylindric space,
$u,u^{\prime}\in\tau^{\mathcal{C}}$ be clopen sets on it, then

  1. 1.

If $j\not\in\Delta(u)$, then $u(\frac{i}{j})=u$;

  2. 2.

If $i\neq j$, then $(-u)(\frac{i}{j})=-(u(\frac{i}{j}))$ and $(u\cap
u^{\prime})(\frac{i}{j})=u(\frac{i}{j})\cap u^{\prime}(\frac{i}{j})$;

  3. 3.

If $j_{1}\neq j_{2}$ and $\\{i_{i},i_{2}\\}\cap\\{j_{1},j_{2}\\}=\emptyset$,
then
$u(\frac{i_{1}}{j_{1}})(\frac{i_{2}}{j_{2}})=u(\frac{i_{2}}{j_{2}})(\frac{i_{1}}{j_{1}})$;

  4. 4.

If $k\not\in\Delta(u),i\not\in\\{j,k\\}$, then
$u(\frac{k}{j})(\frac{i}{k})=u(\frac{i}{j})$;

  5. 5.

If $j\not\in\Delta(u),i\neq j$, then $(u\cap
D_{ij})(\frac{i}{j})=u(\frac{j}{i})(\frac{i}{j})=u$;

  6. 6.

If no duplicate element in $\\{j_{i}:1\geq i\geq n\\}$, for any
$r,s\in\alpha$, $k_{r}=k_{s}\Leftrightarrow
k^{\prime}_{r}=k^{\prime}_{s}\Rightarrow i_{r}=i_{s}$ and
$(\\{k_{1},...k_{n}\\}\cup\\{k^{\prime}_{1},...k^{\prime}_{n}\\})\cap(\\{i_{1},...,i_{n}\\}\cup\\{j_{1},...,j_{n}\\})=\emptyset$
and $\Delta(u)\subseteq\\{j_{1},...,j_{n}\\}$, then
$u(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{i_{1}}{k_{1}})...(\frac{i_{n}}{k_{n}})=u(\frac{k^{\prime}_{1}}{j_{1}})...(\frac{k^{\prime}_{n}}{j_{n}})(\frac{i_{1}}{k^{\prime}_{1}})...(\frac{i_{n}}{k^{\prime}_{n}})$.

Proof

  1. 1.

For any $a\in|\mathcal{C}|$,

| $a\in u(\frac{i}{j})$
---|---
$\Leftrightarrow$ |  $\exists b\sim_{j}a$ s.t. $b\in u\cap D^{\mathcal{C}}_{ij}$
$\Leftrightarrow$ |  $\exists b\sim_{j}a$ s.t. $b\in[u]_{j}\cap D^{\mathcal{C}}_{ij}$
$\Leftrightarrow$ |  $a\in[u]_{j}$ and $\exists b\sim_{j}a$ s.t. $b\in[u]_{j}\cap D^{\mathcal{C}}_{ij}$
$\Leftrightarrow$ |  $a\in u$ and $\exists b\sim_{j}a$ s.t. $b\in[u]_{j}\cap D^{\mathcal{C}}_{ij}$
$\Leftrightarrow$ |  $a\in u$ ($[u]_{j}\cap D^{\mathcal{C}}_{ij}\supseteq[a]_{j}\cap D^{\mathcal{C}}_{ij}\neq\emptyset$)

  2. 2.

For any $a\in|\mathcal{C}|$,

| $a\in(-u)(\frac{i}{j})$
---|---
$\Leftrightarrow$ | $[a]_{j}\cap(-u\cap D^{\mathcal{C}}_{ij})\neq\emptyset$
$\Leftrightarrow$ |  $[a]_{j}\cap(u\cap D_{ij})=\emptyset\quad$(because $\|[a]_{j}\cap D^{\mathcal{C}}_{ij}\|=1$)
$\Leftrightarrow$ | $[a]_{j}\subseteq-(u\cap D^{\mathcal{C}}_{ij})$
$\Leftrightarrow$ | $a\in-(u(\frac{i}{j}))$

| $a\in u(\frac{j}{i})\cap u^{\prime}(\frac{j}{i})$
---|---
$\Leftrightarrow$ |  $\exists b,c\sim_{i}a$ s.t. $b\in u\cap D^{\mathcal{C}}_{ij}$ and $c\in u^{\prime}\cap D^{\mathcal{C}}_{ij}$
$\Leftrightarrow$ |  $\exists b,c\sim_{i}a$ s.t. $b\in u\cap D^{\mathcal{C}}_{ij}$ and $c\in u^{\prime}\cap D^{\mathcal{C}}_{ij}$ and $c=b$ (because $\|[b]_{i}\cap D^{\mathcal{C}}_{ij}\|=1$)
$\Leftrightarrow$ |  $\exists b\sim_{i}a$ s.t. $b\in u\cap u^{\prime}\cap D^{\mathcal{C}}_{ij}$
$\Leftrightarrow$ | $a\in(u\cap u^{\prime})(\frac{j}{i})$

  3. 3.

For any $a\in|\mathcal{C}|$,

| $a\in u(\frac{i_{1}}{j_{1}})(\frac{i_{2}}{j_{2}})$
---|---
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{j_{1}}b\sim_{j_{2}}a$ and $c\in u\cap D^{\mathcal{C}}_{i_{1}j_{1}},b\in D^{\mathcal{C}}_{i_{2}j_{2}}$
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{j_{1}}b\sim_{j_{2}}a$ and $c\in u\cap D^{\mathcal{C}}_{i_{1}j_{1}},b,c\in D^{\mathcal{C}}_{i_{2}j_{2}}$ (because $D^{\mathcal{C}}_{i_{2}j_{2}}$ is $\sim_{j_{1}}$-saturated)
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{j_{1}}b\sim_{j_{2}}a$ and $c\in u\cap D^{\mathcal{C}}_{i_{1}j_{1}}\cap D^{\mathcal{C}}_{i_{2}j_{2}}\quad$(similarly)
$\Leftrightarrow$ | $a\in[[u\cap D^{\mathcal{C}}_{i_{1}j_{1}}\cap D^{\mathcal{C}}_{i_{2}j_{2}}]_{j_{1}}]_{j_{2}}$
$\Leftrightarrow$ | $a\in[[u\cap D^{\mathcal{C}}_{i_{1}j_{1}}\cap D^{\mathcal{C}}_{i_{2}j_{2}}]_{j_{2}}]_{j_{1}}$
$\Leftrightarrow$ | $a\in u(\frac{i_{2}}{j_{2}})(\frac{i_{1}}{j_{1}})$

  4. 4.

It is obviously true when $j=k$. Consider the case when $j\neq k$: for any
$a\in|\mathcal{C}|$,

| $a\in u(\frac{k}{j})(\frac{i}{k})$
---|---
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{j}b\sim_{k}a$ and $c\in u\cap D^{\mathcal{C}}_{kj},b\in D^{\mathcal{C}}_{ik}$
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{j}b\sim_{k}a$ and $c\in u\cap D^{\mathcal{C}}_{kj},b,c\in D^{\mathcal{C}}_{ik}$ (because $D^{\mathcal{C}}_{ik}$ is $\sim_{j}$-saturated)
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{j}b\sim_{k}a$ and $c\in u\cap D^{\mathcal{C}}_{kj}\cap D^{\mathcal{C}}_{ik}\quad$(similarly)
$\Leftrightarrow$ | $a\in[[u\cap D^{\mathcal{C}}_{kj}\cap D^{\mathcal{C}}_{ik}]_{j}]_{k}$
$\Leftrightarrow$ | $a\in[[u\cap D^{\mathcal{C}}_{kj}\cap D^{\mathcal{C}}_{ik}]_{k}]_{j}$
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{k}b\sim_{j}a$ and $c\in u\cap D^{\mathcal{C}}_{kj}\cap D^{\mathcal{C}}_{ik}$
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{k}b\sim_{j}a$ and $c\in u\cap D^{\mathcal{C}}_{kj}\cap D^{\mathcal{C}}_{ik}$ and $b\in u$ (because $u$ is $\sim_{k}$-saturated)
$\Leftrightarrow$ |  $\exists b,c$ s.t. $c\sim_{k}b\sim_{j}a$ and $c\in D^{\mathcal{C}}_{kj}\cap D^{\mathcal{C}}_{ik}$ and $b\in u\quad$(similarly)
$\Leftrightarrow$ |  $\exists b\sim_{j}a$ s.t. $b\in D^{\mathcal{C}}_{ij}=[D^{\mathcal{C}}_{kj}\cap D^{\mathcal{C}}_{ik}]_{k}$ and $b\in u$
$\Leftrightarrow$ | $a\in u(\frac{i}{j})$

  5. 5.

For any $a\in|\mathcal{C}|$,

| $a\in u(\frac{j}{i})(\frac{i}{j})$
---|---
$\Leftrightarrow$ | $a\in[[u\cap D^{\mathcal{C}}_{ij}]_{i}\cap D^{\mathcal{C}}_{ij}]_{j}$
$\Leftrightarrow$ |  $\exists b\sim_{j}a$ s.t. $b\in D^{\mathcal{C}}_{ij}$ and $\exists c\sim_{i}b$ s.t. $c\in u\cap D^{\mathcal{C}}_{ij}$
$\Leftrightarrow$ |  $\exists b\sim_{j}a$ s.t. $b\in D^{\mathcal{C}}_{ij}$ and $\exists c\sim_{i}b$ s.t. $c\in u\cap D^{\mathcal{C}}_{ij}$ and $c=b$ (because $\|[b]_{i}\cap D^{\mathcal{C}}_{ij}\|=1$)
$\Leftrightarrow$ | $a\in[u\cap D^{\mathcal{C}}_{ij}]_{j}=[(u\cap D^{\mathcal{C}}_{ij})\cap D^{\mathcal{C}}_{ij}]_{j}=(u\cap D^{\mathcal{C}}_{ij})(\frac{i}{j})$
$\Leftrightarrow$ | $[a]_{j}\cap(u\cap D^{\mathcal{C}}_{ij})\neq\emptyset$
$\Leftrightarrow$ |  $[a]_{j}\subseteq u$ and $[a]_{j}\cap D^{\mathcal{C}}_{ij}\neq\emptyset\quad$(because $u$ is $\sim_{j}$-saturated)
$\Leftrightarrow$ | $a\in u$

  6. 6.

WLOG, let $\\{k_{i}:1\geq i\geq n\\}\cap\\{k^{\prime}_{i}:1\geq i\geq
n\\}=\emptyset$, we have: (the following proof will frequently use the
antecedents of the state but will not be indicated in the corresponding place,
please note when reading)

|
$u(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{i_{1}}{k_{1}})...(\frac{i_{n}}{k_{n}})$
---|---
$=$ |  $u(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{k^{\prime}_{1}}{k_{1}})(\frac{k_{1}}{k^{\prime}_{1}})(\frac{i_{1}}{k_{1}})...(\frac{i_{n}}{k_{n}})$(by Proposition 3.1.(5))
| ……
$=$ | $u(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{k^{\prime}_{1}}{k_{1}})...(\frac{k^{\prime}_{s}}{k_{s}})(\frac{k_{s}}{k^{\prime}_{s}})...(\frac{k_{1}}{k^{\prime}_{1}})(\frac{i_{1}}{k_{1}})...(\frac{i_{n}}{k_{n}})$
| (if there is $r<s,k_{r}=k_{s}$, by Proposition 3.1.(1),
$(\frac{k^{\prime}_{s}}{k_{s}})$ can be added
| after $(\frac{k^{\prime}_{r}}{k_{r}})$, and $(\frac{k_{r}}{k^{\prime}_{r}})$
can be recorded as $(\frac{k_{s}}{k^{\prime}_{s}})$ and
$(\frac{k_{r}}{k^{\prime}_{r}})$ can be added
| after it, and then $(\frac{k^{\prime}_{s}}{k_{s}})$ and
$(\frac{k_{s}}{k^{\prime}_{s}})$ are moved to the desired position by
| the Proposition 3.1.(3); if there is no $r<s$, using the Proposition
3.1.(5))
| $\dots\dots$
$=$ |  $u(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{k^{\prime}_{1}}{k_{1}})...(\frac{k^{\prime}_{n}}{k_{n}})(\frac{k_{n}}{k^{\prime}_{n}})...(\frac{k_{1}}{k^{\prime}_{1}})(\frac{i_{1}}{k_{1}})...(\frac{i_{n}}{k_{n}})$ (A)
$=$ | $u(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{k^{\prime}_{s_{11}}}{k_{s_{11}}})(\frac{k^{\prime}_{s_{12}}}{k_{s_{12}}})...(\frac{k^{\prime}_{s_{1t_{1}}}}{k_{s_{1t_{1}}}})...(\frac{k^{\prime}_{ht_{h}}}{k_{ht_{h}}})(\frac{k_{n}}{k^{\prime}_{n}})...(\frac{k_{2}}{k^{\prime}_{2}})(\frac{i_{1}}{k^{\prime}_{1}})(\frac{i_{2}}{k_{2}})...(\frac{i_{n}}{k_{n}})$
| (here
$(\frac{k^{\prime}_{s_{11}}}{k_{s_{11}}}),...,(\frac{k^{\prime}_{ht_{h}}}{k_{ht_{h}}})$
lists all the $(\frac{k^{\prime}_{s}}{k_{s}})$ in the previous equation, and
| for each $l\leq h$,
$(\frac{k^{\prime}_{l2}}{k_{l2}}),...,(\frac{k^{\prime}_{lt_{l}}}{k_{lt_{l}}})$
lists all the $(\frac{k^{\prime}_{s}}{k_{s}})$ with the same shape as
|  $(\frac{k^{\prime}_{l1}}{k_{l1}})$, by Proposition3.1.(3), two equations
are equal)
$=$ | $u(\frac{k_{1}}{j_{1}})...(\frac{k_{n-1}}{j_{n-1}})(\frac{k^{\prime}_{s_{11}}}{k_{s_{11}}})...(\frac{k^{\prime}_{s_{1t_{1}}}}{k_{s_{1t^{\prime}_{1}}}})(\frac{k^{*}_{n}}{j_{n}})(\frac{k^{\prime}_{s_{21}}}{k_{s_{21}}})...$
| (if $k_{n}\neq k_{s_{11}}$, then repeated use of Proposition 3.1.(3) yields
this equ-
| -ation, and we know $t_{1}=t^{\prime}_{1},k^{*}_{n}=k_{n}$; if
$k_{n}=k_{s_{11}}$, then by Proposition
|  3.1.(4), $(\frac{k_{n}}{j_{n}})(\frac{k^{\prime}_{s_{11}}}{k_{s_{11}}})$
can be synthesized into $(\frac{k^{\prime}_{n}}{j_{n}})$ then by Proposition
|  3.1.(3)
$(\frac{k^{\prime}_{s_{12}}}{k_{s_{12}}})...(\frac{k^{\prime}_{s_{1t_{1}}}}{k_{s_{1t^{\prime}_{1}}}})$
can be moved before $(\frac{k^{\prime}_{n}}{j_{n}})$ and therefore this eq-
| -uation can be obtained, and we know $t^{\prime}_{1}=t_{1}-1$,
$k^{*}_{n}=k^{\prime}_{n}$) (B)
| $\dots\dots$
$=$ |  $u(\frac{k^{\prime}_{1}}{j_{1}})...(\frac{k^{\prime}_{n}}{j_{n}})(\frac{k_{n}}{k^{\prime}_{n}})...(\frac{k_{2}}{k^{\prime}_{2}})(\frac{i_{2}}{k_{2}})...(\frac{i_{n}}{k_{n}})\quad$(C)
| (In the same way as steps A-B, repeated several times to obtain)
| $\dots\dots$
$=$ |  $u(\frac{k^{\prime}_{1}}{j_{1}})...(\frac{k^{\prime}_{n}}{j_{n}})(\frac{i_{1}}{k^{\prime}_{1}})...(\frac{i_{n}}{k^{\prime}_{n}})\quad$(In the same way as steps A-C)

$\Box$

The following definition is well-defined by Proposition 3.1.

###### Definition 3.2

Let $\mathcal{C}_{\alpha}$ be a basis-finite cylindric space,
$\rho:\alpha^{\alpha}$ be a (partial) mapping,

  * •

Let $B$ be a cylindric basis of $\mathcal{C}$, for any $u\in B$ with
$\Delta(u)\subseteq dom(\rho)$, we write $\rho
u:=u(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{n})}{k_{n}})$
s.t. no duplicate elements in both $\\{k_{i}:1\geq i\geq n\\}$ and
$\\{j_{i}:1\geq i\geq n\\}$, $\Delta(u)\subseteq\\{j_{1},...,j_{n}\\}$ and
$\\{k_{1},...k_{n}\\}\cap(\\{j_{1},...,j_{n}\\}\cup\\{\rho(j_{1},...,\rho(j_{n}))\\})=\emptyset$.
We call $\rho u$ a permutation of $u$;

  * •

For a closed set $u$ with $\Delta(u)\subseteq dom(\rho)$, we write $\rho
u:=\bigcap\\{\rho v:v\in B\mbox{ and }u\subseteq v\\}$, call $\rho u$ a
permutation of $u$; the permutation of open sets also can be defined in the
same way.

Now we define a cylindric space that is very important in the following text:

###### Definition 3.3

We call a Stone basis-finite cylindric space an FOL space.

For FOL space, the following proposition holds:

###### Theorem 3.1

Let $\mathcal{C}_{\alpha}$ be an FOL space, $\rho:\alpha^{\alpha}$ be a
mapping, $a\in|\mathcal{C}_{\alpha}|$, we have

  1. 1.

If for any $k\in ran(\rho)$ and $i,j\in\rho^{-1}[k]$, we have $a\in D_{ij}$,
then $\rho\\{a\\}\neq\emptyset$;

  2. 2.

If $\rho$ is a surjection, then $\|\rho\\{a\\}\|\leq 1$;

  3. 3.

$\|\\{a^{\prime}:a\in\rho\\{a^{\prime}\\}\\}\|=1$;

  4. 4.

For closed sets $u,v,w$, if $\rho u=v,\rho^{\prime}v=w$, then
$(\rho^{\prime}\circ\rho)u=w$.

Proof Let $B$ be a cylindric basis of $\mathcal{C}$,

  1. 1.

We know $\rho\\{a\\}=\bigcap\\{\rho u:u$ is a clopen neighborhood of $a\\}$.

For any clopen neighborhood $u_{1},...,u_{m}\ni a$, we know there are
$k_{1},...,k_{n},j_{i},...,j_{n}$, $\rho
u_{i}=u_{i}(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})$
$(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{n})}{k_{n}})$ satisfying the
conditions in Definition 3.2. Let
$(j_{l_{1}},j_{l^{\prime}_{1}}),...,(j_{l_{s}},j_{l^{\prime}_{s}})$ be all
pairs in $\\{j_{1},...,j_{n}\\}$ satisfying
$\rho(j_{l_{r}})=\rho(j_{l_{r^{\prime}}})$, we know
$\bigcap_{r=1}^{s}D^{\mathcal{C}}_{j_{l_{r}},j_{l^{\prime}_{r}}}\cap\bigcap_{i=1}^{n}u_{i}\neq\emptyset$
because $a$ is in it by the assumption.

| $\bigcap_{r=1}^{s}\rho
D^{\mathcal{C}}_{j_{l_{r}},j_{l^{\prime}_{r}}}\cap\bigcap_{i=1}^{n}\rho u_{i}$
---|---
$=$ | $\bigcap_{r=1}^{s}(D^{\mathcal{C}}_{j_{l_{r}},j_{l^{\prime}_{r}}}(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{n})}{k_{n}}))\cap\bigcap_{i=1}^{n}(u_{i}(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{n})}{k_{n}}))$
$=$ |  $(\bigcap_{r=1}^{s}D^{\mathcal{C}}_{k_{l_{r}},k_{l^{\prime}_{r}}}\cap(\bigcap_{i=1}^{n}u_{i})(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}}))(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{n})}{k_{n}})$ (use Proposition 3.1.(2) repeatedly)

Since $\bigcap_{r=1}^{s}D^{\mathcal{C}}_{j_{l_{r}},j_{l^{\prime}_{r}}}$ $\cap$
$\bigcap_{i=1}^{n}u_{i}\neq\emptyset$, we know
$\bigcap_{r=1}^{s}D^{\mathcal{C}}_{k_{l_{r}},k_{l^{\prime}_{r}}}$
$\cap(\bigcap_{i=1}^{n}u_{i})(\frac{k_{1}}{j_{1}})$ $...$
$(\frac{k_{n}}{j_{n}})\neq\emptyset$. For each $i\in\\{0,...,n\\}$, let
$X_{i}:=\bigcap_{r=1}^{s}D^{\mathcal{C}}_{k_{l_{r}},k_{l^{\prime}_{r}}}$
$\cap$ $(\bigcap_{i=1}^{n}u_{i})(\frac{k_{1}}{j_{1}})$ $...$
$(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})$ $...$
$(\frac{\rho(j_{i})}{k_{i}})$. We use induction on $i\in\\{0,...,n\\}$ as
follow to show $X_{i}\neq\emptyset$:

     * •

IH: For $i^{\prime}<i$,
$X_{i^{\prime}}=\bigcap\\{D^{\mathcal{C}}_{\rho(j_{l_{r}}),\rho(j_{l^{\prime}_{r}})}:l_{r},l^{\prime}_{r}\leq
i^{\prime}\\}\cap\bigcap\\{D^{\mathcal{C}}_{\rho(j_{l_{r}}),k_{l^{\prime}_{r}}}:l_{r}\leq
i,l^{\prime}_{r}>i^{\prime}\\}\cap\bigcap\\{D^{\mathcal{C}}_{k_{l_{r}},k_{l^{\prime}_{r}}}:l_{r},l^{\prime}_{r}>i^{\prime}\\}\cap(\bigcap_{t=1}^{n}u_{t})(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{i^{\prime}})}{k_{i^{\prime}}})\neq\emptyset$.

     * •

IS:

       * –

If $\rho(j_{i})$ does not in the dimension of $X_{i-1}$, then
$X_{i}\neq\emptyset$. By Proposition 3.1.(2),
$X_{i}=\bigcap\\{D^{\mathcal{C}}_{\rho(j_{l_{r}}),\rho(j_{l^{\prime}_{r}})}:l_{r},l^{\prime}_{r}\leq
i\\}\cap\bigcap\\{D^{\mathcal{C}}_{\rho(j_{l_{r}}),k_{l^{\prime}_{r}}}:l_{r}\leq
i,l^{\prime}_{r}>i\\}\cap\bigcap\\{D^{\mathcal{C}}_{k_{l_{r}},k_{l^{\prime}_{r}}}:l_{r},l^{\prime}_{r}>i\\}\cap(\bigcap_{t=1}^{n}u_{t})(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{i})}{k_{i}})$;

       * –

If $\rho(j_{i})$ does in the dimension of $X_{i-1}$, then there is
$i^{\prime}<i$ s.t. $\rho(j_{i})=\rho(j_{i^{\prime}})$. By IH, we know
$X_{i-1}\subseteq D^{\mathcal{C}}_{\rho(j_{i^{\prime}}),k_{i}}$, then
$X_{i}=[X_{i-1}\cap D^{\mathcal{C}}_{\rho(j_{i}),k_{i}}]_{k_{i}}\neq\emptyset$
and by Proposition 3.1.(2),
$X_{i}=\bigcap\\{D^{\mathcal{C}}_{\rho(j_{l_{r}}),\rho(j_{l^{\prime}_{r}})}:l_{r},l^{\prime}_{r}\leq
i\\}\cap\bigcap\\{D^{\mathcal{C}}_{\rho(j_{l_{r}}),k_{l^{\prime}_{r}}}:l_{r}\leq
i,l^{\prime}_{r}>i\\}\cap\bigcap\\{D^{\mathcal{C}}_{k_{l_{r}},k_{l^{\prime}_{r}}}:l_{r},l^{\prime}_{r}>i\\}\cap(\bigcap_{t=1}^{n}u_{t})(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{i})}{k_{i}})$.

Hence, $\bigcap_{r=1}^{s}\rho
D^{\mathcal{C}}_{j_{l_{r}},j_{l^{\prime}_{r}}}\cap\bigcap_{i=1}^{n}\rho
u_{i}\neq\emptyset$, then we know $\\{\rho u:u$ is a clopen neighborhood of
$a\\}$ has finite intersection property, then by compactness,
$\rho\\{a\\}=\bigcap\\{\rho u:u$ is clopen neighborhood of
$a\\}\neq\emptyset$.

  2. 2.

Assume there are $b,b^{\prime}\in\rho\\{a\\}$ s.t. $b\neq b^{\prime}$, we know
there is a $u\in B$ s.t. $u\ni b,-u\ni b^{\prime}$. Let
$\Delta(u)=\Delta(-u)=\\{x_{1},...,x_{n}\\}$. Since $\rho$ is a surjection, we
know there are $y_{1},..,y_{n}\in\alpha$ s.t. $\rho(y_{i})=x_{i}$. Choose
arbitrary
$k_{1},...,k_{n}\in\alpha\backslash\\{x_{1},...,x_{n},y_{1},...,y_{n}\\}$ and
let
$v=u(\frac{k_{1}}{x_{1}})...(\frac{k_{n}}{x_{n}})(\frac{y_{1}}{k_{1}})...(\frac{y_{n}}{k_{n}})$.
By Proposition 3.1, we know $\rho
v=u(\frac{k_{1}}{x_{1}})...(\frac{k_{n}}{x_{n}})(\frac{y_{1}}{k_{1}})...(\frac{y_{n}}{k_{n}})(\frac{k_{1}}{y_{1}})...(\frac{k_{n}}{y_{n}})(\frac{x_{1}}{k_{1}})...(\frac{x_{n}}{k_{n}})=u,\rho-v=-u(\frac{k_{1}}{x_{1}})...(\frac{k_{n}}{x_{n}})(\frac{y_{1}}{k_{1}})...(\frac{y_{n}}{k_{n}})$
$(\frac{k_{1}}{y_{1}})...(\frac{k_{n}}{y_{n}})$
$(\frac{x_{1}}{k_{1}})...(\frac{x_{n}}{k_{n}})=-u$. Clearly $a\in v$ or
$a\in-v$, then $\rho\\{a\\}\subseteq u$ or $\rho\\{a\\}\subseteq-u$, which
contradicts the requirement for $u$.

  3. 3.

Let $U=\\{u\in B:a\in\rho u\\}$, we know
$\\{a^{\prime}:a\in\rho\\{a^{\prime}\\}\\}=\bigcap U$. Choose an arbitrary
finite $V\subseteq U$, by Proposition 3.1.(2), $\rho(\bigcap V)=\bigcap\\{\rho
v:v\in V\\}\ni a$, then $\bigcap V\neq\emptyset$. By compactness, $\bigcap
U\neq\emptyset$.

For any $u\in B$, if $u\not\in U$, then $a\not\in\rho u$, then by Proposition
3.1.(2), $a\in-\rho u=\rho-u$, then $-u\in U$, then by $T_{2}$ property,
$\|\bigcap U\|=1$.

  4. 4.

| $(\rho^{\prime}\circ\rho)u$
---|---
$=$ | $(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho^{\prime}\circ\rho(j_{1}))}{k_{1}})...(\frac{\rho^{\prime}\circ\rho(j_{n}))}{k_{n}})$
$=$ | $(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{k^{\prime}_{n}}{k_{n}})...(\frac{k^{\prime}_{1}}{k_{1}})(\frac{\rho^{\prime}\circ\rho(j_{1}))}{k^{\prime}_{1}})...(\frac{\rho^{\prime}\circ\rho(j_{n}))}{k^{\prime}_{n}})$
| (In the same way as steps A-C of the proof of the Proposition 3.1.(6))
$=$ |  $(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{k^{\prime}_{1}}{k_{1}})...(\frac{k^{\prime}_{n}}{k_{n}})(\frac{\rho^{\prime}\circ\rho(j_{1}))}{k^{\prime}_{1}})...(\frac{\rho^{\prime}\circ\rho(j_{n}))}{k^{\prime}_{n}})$ (Repeated use of the
| Proposition 3.1.(3))
$=$ | $(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{n})}{k_{n}})(\frac{k^{\prime}_{n}}{\rho(j_{n})})...(\frac{k^{\prime}_{1}}{\rho(j_{1})})(\frac{\rho^{\prime}(\rho(j_{1}))}{k^{\prime}_{1}})...(\frac{\rho^{\prime}(\rho(j_{n}))}{k^{\prime}_{n}})$
| (In the same way as steps A-C of the proof of the Proposition 3.1.(6))
$=$ | $(\frac{k_{1}}{j_{1}})...(\frac{k_{n}}{j_{n}})(\frac{\rho(j_{1})}{k_{1}})...(\frac{\rho(j_{n})}{k_{n}})(\frac{k^{\prime}_{1}}{\rho(j_{1})})...(\frac{k^{\prime}_{n}}{\rho(j_{n})})(\frac{\rho^{\prime}(\rho(j_{1}))}{k^{\prime}_{1}})...(\frac{\rho^{\prime}(\rho(j_{n}))}{k^{\prime}_{n}})$
| (Repeated use of the Proposition 3.1.(3))
$=$ | $\rho^{\prime}(\rho u)\quad$
| (By Proposition 3.1.(1), we can add or delete operations
$(\frac{k}{i}),(\frac{\rho^{\prime}(i)}{k})$
| to a format consistent with the definition)
$=$ | $w$

$\Box$

###### Definition 3.4

let $\mathcal{C}_{\alpha}$ be a basis-finite cylindric space,
$s\subseteq\alpha$, $u$ be a closed set in $\mathcal{C}_{\alpha}$ with
dimension $s$ s.t. for any closed set $v$, $\Delta(v)\subseteq s\Rightarrow
u\subseteq v$ or $u\subseteq-v$. We call $u$ a complete closed set of
dimension $s$. For $a\in|\mathcal{C}|$, write the complete closed set of
dimension $s$ containing $a$ as $a|_{s}$.

We can now define the permutation relation between points based on the above
theorem.

###### Definition 3.5

Let $\mathcal{C}_{\alpha}$ be an FOL space.

  * •

For $a,b\in|\mathcal{C}_{\alpha}|$ and (partial) mapping
$\rho:\alpha^{\alpha}$, if $b\in\rho\\{a|_{dom(\rho)}\\}$, then we say that
$a$ is a (partial) factor of $b$, and denote as $\rho^{-1}b=a$,
$a\prec_{\rho}b$ or $a\prec b$;

  * •

For $a,b\in|\mathcal{C}_{\alpha}|$ and $\rho:\alpha^{\alpha}$, if $\rho$ is a
surjection and $\rho^{-1}b=a$ (knowing that at this time,
$\rho\\{a\\}=\\{b\\}$), we say that $b$ is a permutation of $a$, and denote as
$\rho a=b$.

###### Example 3.1

Let $\mathfrak{A},\mathfrak{B}$ be two countable first-order structures s.t.
there is a (partial) elementary embedding
$f:\mathfrak{A}\rightarrow\mathfrak{B}$. Then for
$a\in\mathcal{C}^{\mathfrak{A}}_{\omega}$,
$b\in\mathcal{C}^{\mathfrak{B}}_{\omega}$ being domain points and
$a^{\prime},b^{\prime}\in\mathcal{C}_{T}$ realized by $a,b$ respectively,
$a^{\prime}$ is a (partial) factor of $b^{\prime}$. If $f$ is a isomorphism
and sequence $b$ has no duplicates, then $b^{\prime}$ is a permutation of
$a^{\prime}$.

###### Definition 3.6

Let $\mathcal{C}_{\alpha}$ be an FOL space, $a,b\in|\mathcal{C}|$. If there is
a $\rho:a\prec b$ s.t. for any $j\in\alpha\backslash ran(\rho)$, there is an
$i\in ran(\rho)$ s.t. $b\in D_{ij}$, then we say $b$ is an equivalent point of
$a$ and denote this as $\rho:a\asymp b$.

###### Proposition 3.2

Let $\mathcal{C}_{\alpha}$ be an FOL space, the relation $\asymp$ is an
equivalence relation on $|\mathcal{C}|$:

  * •

$a\asymp b\Leftrightarrow$ there is a $\rho:a\prec b$ s.t. for any
$j\in\alpha\backslash ran(\rho)$, there is an $i\in ran(\rho)$ s.t. $b\in
D_{ij}$ (denote this as $\rho:a\asymp b$).

Proof

  * •

Reflexivity: Let $\rho$ be the identity, clearly $\rho:a\asymp a$.

  * •

Symmetry: Assume $\rho:a\asymp b$, we define $\rho^{\prime}:\alpha^{\alpha}$
as $i\mapsto min\\{i^{\prime}:b\in D_{\rho(i^{\prime})i}\\}$, it’s easy to see
$\rho^{\prime-1}a=b$.

For any $j\in\alpha\backslash ran(\rho^{\prime})$, consider
$j^{\prime}=min\\{k:\rho(k)=\rho(j)\\}$, we know
$j^{\prime}=\rho^{\prime}(\rho(j))\in ran(\rho^{\prime})$ and $a\in
D_{jj^{\prime}}$. Hence $\rho^{\prime}:b\asymp a$.

  * •

Transitivity: Assume $\rho:a\asymp b$,, $\rho^{\prime}:b\asymp c$, then by
Theorem 3.1.(4), we know $\rho^{\prime}\circ\rho\\{a\\}\supseteq\\{c\\}$ i.e.
$\rho^{\prime}\circ\rho:a\prec c$. For any $i\in\alpha\backslash
ran(\rho^{\prime}\circ\rho)$, we know there is an $i^{\prime}\in
ran(\rho^{\prime})$ s.t. $c\in D_{ii^{\prime}}$. If $i^{\prime}\not\in
ran(\rho^{\prime}\circ\rho)$, then
$\rho^{\prime-1}[i^{\prime}]\subseteq\alpha\backslash ran(\rho)$, then for any
$i^{+}\in\rho^{\prime-1}[i^{\prime}]$ there is an $i^{*}\in ran(\rho)$ s.t.
$b\in D_{i^{+}i^{*}}$, then $c\in D_{i^{\prime}\rho^{\prime}(i^{*})}$, then
$c\in D_{i\rho^{\prime}(i^{*})}$, then clearly we have
$\rho^{\prime}(i^{*})\in ran(\rho^{\prime}\circ\rho)$. Hence
$\rho^{\prime}\circ\rho:a\asymp c$.

$\Box$

###### Example 3.2

Let $a,b,a,^{\prime}b^{\prime}$ be as in Example 3.1, $f$ be a isomorphism,
then $a^{\prime}\asymp b^{\prime}$ regardless of whether $b^{\prime}$ has
duplicates.

###  3.2 $\alpha$-expansion

In this subsection, we will study how to “expand" a basis-finite cylindric
space into an FOL space of a specific dimension and further construct a one-
to-one correspondence between FOL spaces of different dimensions. First, for a
basis-finite cylindric space $\mathcal{C}_{\beta}$ and an ordinal
$\alpha\geq\beta$, consider the following construction:

Let $B$ be the family of all clopen sets in $\mathcal{C}_{\alpha}$, $M$ be the
collection of all mappings from $\beta$ to $\alpha$. We call $x\subseteq
M\times B$ an atom if it satisfies the following conditions:

  1. 1.

For any $\rho\in M,u\in B$, $(\rho,u)\in x\Leftrightarrow(\rho,-u)\not\in x$;

  2. 2.

For any $\rho\in M,u,v\in B$, if $u\subseteq v$ and $(\rho,u)\in x$, then
$(\rho,v)\in x$;

  3. 3.

For any $\rho\in M,u,v\in B$, if $u\cap v\neq\emptyset$ and
$(\rho,u),(\rho,v)\in x$, then $(\rho,u\cap v)\in x$;

  4. 4.

For any $\rho,\mu,\sigma\in M,u,v\in B$, if
$\mu|_{\Delta(u)}=\sigma|_{\Delta(u)}$, $\rho[\Delta(u)]\subseteq\beta$ and
$\rho v=u$, then $(\mu\circ\rho,v)\in x\Leftrightarrow(\sigma,u)\in x$; 444For
any mapping $\rho:\alpha^{\beta}$, any $u\in B$ satisfying
$\rho[\Delta(u)]\subseteq\beta$, clearly there is
$\rho^{\prime}:\beta^{\beta}$ s.t.
$\rho^{\prime}|_{\Delta(u)}=\rho|_{\Delta(u)}$, we write $\rho
u:=\rho^{\prime}u$.

Before starting the formal construction, we still need to verify one property.

###### Proposition 3.3

Define $X_{\mathcal{C}}:=\\{a\subseteq M\times B:a$ is atom$\\}$. For any
$(\rho,u)\in M\times B$, there is an atom $x\in X_{\mathcal{C}}$ s.t.
$(\rho,u)\in x$ and therefore $X_{\mathcal{C}}\neq\emptyset$.

Proof Well-order $M\times B$ with $(\rho,u)$ as the least element and let its
isomorphic ordinal be $\gamma$. For each $i\in\gamma$, write the $i$-th
element as $(\rho_{i},u_{i})$. We use transfinite induction to define the set
$x_{i}$ for each $i\in\gamma$ and verify that they satisfy the following
conditions:

  1. 1.

For any $\mu\in M$, any finite $U\subseteq B$, if $\\{(\mu,v):v\in
U\\}\subseteq x_{i}$, then for any $w\subseteq-\bigcap U$, $(\mu,w)\not\in
x_{i}$;

  2. 2.

It satisfies the fourth clause of the definition of the atom.

The following proof will refer to these two conditions as inductive
conditions.

BS: $x_{0}:=\\{(\rho\circ\mu,v):\mu[\Delta(v)]=\Delta(u)\mbox{ and }\mu
v=u\\}$. clearly, $x_{0}$ satisfies the inductive conditions;

IH: For $j<i$, $x_{j}$ has been constructed and satisfies the induction
conditions, and for $j^{\prime}<j<i$, $x_{j^{\prime}}\subseteq x_{j}$; here we
prove a corollary of IH, namely $\bigcup_{j<i}x_{j}$ satisfies the induction
conditions:

It is easy to see that $\bigcup_{j<i}x_{j}$ satisfies the fourth clause of the
definition of the atom. If there is $\mu\in M$ and finite $U^{+}\subseteq B$
s.t. $\\{(\mu,v):v\in U^{+}\\}\subseteq\bigcup_{j<i}x_{j}$ constitutes a
counterexample to the first clause of the inductive conditions, then since
$U^{+}$ is finite, we know there is $j<i$ s.t. $\\{(\mu,v):v\in
U^{+}\\}\subseteq x_{j}$, then $x_{j}$ does not satisfy the inductive
conditions, which contradicts IH.

IS: If $(\rho_{i},u_{i})\in\bigcup_{j<i}x_{j}$, then let
$x_{i}:=\bigcup_{j<i}x_{j}$. We know $x_{i}$ satisfies the inductive
conditions by IH.

If $(\rho_{i},u_{i})\not\in\bigcup_{j<i}x_{j}$,

  * •

If there is a finite $U\subseteq B$ s.t. $\\{(\rho_{i},v):v\in
U\\}\subseteq\bigcup_{j<i}x_{j}$ and $u_{i}\subseteq-\bigcap U$, then let
$x_{i}:=\bigcup_{j<i}x_{j}\cup\\{(\rho_{i}\circ\mu,v):\mu[\Delta(v)]=\Delta(u_{i})\mbox{
and }\mu v=-u_{i}\\}$, It’s easy to see that it satisfies the fourth clause of
definition of atom. Assume there is a $\mu\in M$, finite $U^{+}\subseteq B$
s.t. $\\{(\mu,v):v\in U^{+}\\}\subseteq x_{i}$ constituting a counterexample
to the first clause of the inductive conditions. By the corollary of IH,
$\\{(\mu,v):v\in U^{+}\\}\subseteq x_{i}\not\subseteq\bigcup_{j<i}x_{j}$. By
the fourth clause of definition of atom, there is a finite $V^{+}\subseteq B$
s.t. $\\{(\rho_{i},v):v\in V^{+}\\}\subseteq x_{i}$ constitutes a
counterexample to the first clause of the inductive conditions, and
$(\rho_{i},-u_{i})\in\\{(\rho_{i},v):v\in
V^{+}\\}\subseteq\\{(\rho_{i},-u_{i})\\}\cup\bigcup_{j<i}x_{j}$. By the
construction method, $\\{(\rho_{i},w):w\in U\cup
V^{+}\backslash\\{-u_{i}\\}\\}$ constitutes a counterexample to the first
clause of the inductive conditions and clearly $\\{(\rho_{i},w):w\in U\cup
V^{+}\backslash\\{-u_{i}\\}\\}\subseteq\bigcup_{j<i}x_{j}$ which contradicts
the corollary of IH;

  * •

If there is no such $U$, let
$x_{i}:=\bigcup_{j<i}x_{j}\cup\\{(\rho_{i}\circ\mu,v):\mu[\Delta(v)]=\Delta(u_{i})\mbox{
and }\mu v=u_{i}\\}$, then the inductive conditions is satisfied in the same
way as the previous case.

Let $x=\bigcup_{i<\gamma}x_{i}$. By the same reasoning as the corollary of IH,
it can be obtained that $x$ satisfies the induction conditions. Then we have

  * •

$x$ obviously satisfies the first and the fourth clauses of the definition of
the atom;

  * •

For any $\rho\in M,u,v,w\in B$ satisfying $u\cap v\neq\emptyset$,
$(\rho,u),(\rho,v)\in x$ and $u\cap v\subseteq w$, assume $(\rho,w)\not\in x$,
we know $(\rho,-w)\in x$, then $\\{(\rho,u),(\rho,v),(\rho,-w)\\}$ constitutes
a counterexample to the inductive conditions, contradiction. Hence
$(\rho,w)\in x$. Hence $x$ satisfies the second and the third clauses of the
definition of the atom.

Hence, $x$ is an atom. $\Box$

As the reader may have realized, we want to construct an “ultrafilter space"
of $\mathcal{C}$ and make its dimension $\beta$, each atom being an
“ultrafilter" point in this space. Then, the clopen sets in this space are
naturally defined as follows: for each $(\rho,u)\in M\times B$, define
$X_{(\rho,u)}=\\{x\in X_{\mathcal{C}}:(\rho,u)\in x\\}$.

###### Lemma 3.1

For any $(\rho,u),(\mu,v)\in M\times B$, there is a $\rho^{\prime}\in M$,
$u^{\prime},v^{\prime}\in B$ s.t.
$X_{(\rho,u)}=X_{(\rho^{\prime},u^{\prime})}$,
$X_{(\mu,v)}=X_{(\rho^{\prime},v^{\prime})}$.

Proof obviously there is $\eta\in\beta^{\beta},u^{\prime}\in B$ s.t. $\eta
u^{\prime}=u$ and $\Delta(u^{\prime})\cap\Delta(v)=\emptyset$. We know
$X_{(\rho\circ\eta,u^{\prime})}=X_{(\rho,u)}$. Let $\rho^{\prime}\in M$
contain $(\rho\circ\eta)|_{\Delta(u^{\prime})}$ and $\mu|_{\Delta(v)}$, we
know
$X_{(\rho,u)}=X_{(\rho^{\prime},u^{\prime})},X_{(\mu,v)}=X_{(\rho^{\prime},v})$.
$\Box$

###### Proposition 3.4

$B^{X}:=\\{X_{(\rho,u)}:(\rho,u)\in M\times B\\}$ is a basis of Stone topology
on $X_{\mathcal{C}}$. We write the induced Stone topology as $\tau^{X}$.

Proof For any $(\rho,u),(\mu,v)\in M\times B$ s.t. $X_{(\rho,u)}\cap
X_{(\mu,v)}\neq\emptyset$, by Lemma 3.1, there is $\rho^{\prime}\in M$,
$u^{\prime},v^{\prime}\in B$ s.t.
$X_{(\rho,u)}=X_{(\rho^{\prime},u^{\prime})}$,
$X_{(\mu,v)}=X_{(\rho^{\prime},v^{\prime})}$. Then by the second clause of
definition of atom, $X_{(\rho,u)}\cap
X_{(\mu,v)}=X_{(\rho^{\prime},u^{\prime}\cap v^{\prime})}$. Hence $B^{X}$ is
closed for taking finite intersection. Apparently, both
$X_{(1,|\mathcal{C}|)}=X_{\mathcal{C}},X_{(1,\emptyset)}=\emptyset$ are in
$B^{X}$555Here $1:\beta\rightarrow\alpha$ in $X_{(1,|\mathcal{C}|)}$ denotes
the inclusion mapping, which is the same later in the text.. Hence $B^{X}$ is
a topological basis.

By the definition, $-X_{(\rho,u)}=X_{(\rho,-u)}$, then sets in $B^{X}$ are all
clopen sets. Hence $(X_{\mathcal{C}},\tau^{X})$ is zero-dimensional.

For any $x\neq y\in X_{\mathcal{C}}$, there must be $(\rho,u)\in M\times B$
s.t. $x\in X_{(\rho,u)},y\not\in X_{(\rho,u)}$. Hence
$(X_{\mathcal{C}},\tau^{X})$ is a $T_{2}$ space.

For any $U\subseteq B^{X}$ with finite intersection property, define
$x_{0}=\\{(\rho,u):X_{(\rho,u)}\in U\\}$. It is easy to see that $x_{0}$
satisfies the inductive conditions in the proof of Proposition 3.3. Then, in
the same way as the proof of Proposition 3.3, $x_{0}$ can be expanded to an
atom $x$ and $x\in\bigcap U$. By the arbitrariness of $U$, we know
$(X_{\mathcal{C}},\tau^{X})$ is compact.

$\Box$

For any $i,j\in\alpha$, we know there are
$i^{\prime},j^{\prime}\in\beta,\rho\in M$ s.t.
$\rho(i^{\prime})=i,\rho(j^{\prime})=j$, then define
$D_{ij}=X_{(\rho,D^{\mathcal{C}}_{i^{\prime}j^{\prime}})}$ (It is easy to see
this is well-defined). For any $i\in\alpha$, define relation $\backsim_{i}$ on
$X_{\mathcal{C}}$ as: $x\backsim_{i}y\Leftrightarrow\\{(\rho,u)\in
x:i\not\in\rho[\Delta(u)]\\}=\\{(\rho,u)\in y:i\not\in\rho[\Delta(u)]\\}$.

###### Lemma 3.2

For $(\rho,u),(\mu,v)\in M\times B$, $i\in\beta$ satisfying
$\mu(i)\not\in\rho[\Delta(u)]$, we have $X_{(\rho,u)}\cap
X_{(\mu,[v]_{i})}\neq\emptyset\Rightarrow X_{(\rho,u)}\cap
X_{(\mu,v)}\neq\emptyset$.

Proof By Lemma 3.1, w.l.o.g. let $\rho=\mu$, then we have $i\not\in\Delta(u)$.
If $X_{(\rho,u)}\cap X_{(\mu,v)}=X_{(\rho,u\cap v)}=\emptyset$, then by
definition, $u\cap v=\emptyset$. then $u\cap[v]_{i}=[u\cap
v]_{i}=[\emptyset]_{i}=\emptyset$, then $X_{(\rho,u)}\cap
X_{(\mu,[v]_{i})}=X_{(\rho,u\cap[v]_{i})}=\emptyset$. $\Box$

###### Theorem 3.2

Let $\mathcal{C}_{\beta}$ be a basis-finite cylindric space. For
$\alpha\geq\beta$, define
$\mathcal{C}^{\alpha}=((X_{\mathcal{C}},\tau^{X}),\\{\backsim_{i}:i\in\alpha\\},\\{D_{ij}:i,j\in\alpha\\})$,
then

  1. 1.

$\mathcal{C}^{\alpha}$ forms an FOL space;

  2. 2.

If $\alpha=\beta$, then there is a basis-preserving C-mapping
$f:\mathcal{C}_{\beta}\rightarrow\mathcal{C}^{\alpha}$ which is an injection
if $\mathcal{C}_{\beta}$ is a $T_{2}$ space.

Proof To prove that $\mathcal{C}^{\alpha}$ is an FOL space, by Proposition
3.4, we just need to prove that it is a cylindric space:

  * •

The commutativity of $\\{\backsim_{i}:i\in\alpha\\}$

For $x,y,z\in X_{\mathcal{C}}$, if $x\backsim_{i}z\backsim_{j}y$, let
$x^{ij}:=\\{(\rho,u)\in x:i,j\not\in\rho[\Delta(u)]\\},y^{ij}:=\\{(\rho,u)\in
y:i,j\not\in\rho[\Delta(u)]\\}$, then $x^{ij}=y^{ij}$. Let
$x^{j}=\\{(\rho,u)\in x:j\not\in\rho[\Delta(u)]\\}$, $y^{i}=\\{(\rho,u)\in
y:i\not\in\rho[\Delta(u)]\\}$. Write
$j^{\prime}:=\rho^{-1}(j),i^{\prime}:=\mu^{-1}(i)$, then for any $(\rho,u)\in
x^{j}$ and $(\mu,v)\in y^{i}$, $(\rho,[u]_{i^{\prime}})\in
x^{ij}=y^{ij}\ni(\mu,[v]_{j^{\prime}})$, then $x,y\in
X_{(\rho,[u]_{i^{\prime}})}\cap X_{(\mu,[v]_{j^{\prime}})}\neq\emptyset$.
Using Lemma 3.2 twice, we get $X_{(\rho,u)}\cap X_{(\mu,v)}\neq\emptyset$. By
compactness, we know $\bigcap\\{X_{(\rho,u)}:(\rho,u)\in x^{j}\cup
y^{i}\\}\neq\emptyset$ and for any $z^{\prime}$ in this set,
$x\sim_{j}z^{\prime}\sim_{i}y$.

  * •

$B^{X}$ is a cylindric basis about $\\{\backsim_{i}:i\in\alpha\\}$

By definition, $B^{X}$ is closed under taking intersection, union, complement
and for any $i\in\beta$, $(\rho,u)\in B^{X}$. write
$i^{\prime}:=\rho^{-1}(i)$, we show
$[X_{(\rho,u)}]_{i}=X_{(\rho,[u]_{i^{\prime}})}$:

| $x\in[X_{(\rho,u)}]_{i}$
---|---
$\Leftrightarrow$ |  $\exists y\backsim_{i}x$ s.t. $(\rho,u)\in y$
$\Leftrightarrow$ |  $x^{i}\cup\\{(\rho,u)\\}$ can be expanded to a atom
$\Leftrightarrow$ |  $\\{X_{\rho,u}\\}\cup\\{X_{(\mu,v)}:(\mu,v)\in x^{i}\\}$ has finite intersection property
| (by compactness)
$\Leftrightarrow$ | For any $(\mu,v)\in x^{i},X_{(\rho,u)}\cap X_{(\mu,v)}\neq\emptyset$
$\Leftrightarrow$ | For any $(\mu,v)\in x^{i},X_{(\rho,[u]_{i^{\prime}})}\cap X_{(\mu,v)}\neq\emptyset\quad$
| (by definition of $x^{i}$, $\rho(i^{\prime})=i\not\in\mu(\Delta(v))$, then
by Lemma 3.2 we get this)
$\Leftrightarrow$ | $(\rho,[u]_{i^{\prime}})\in x^{i}\subseteq x\quad$
| ($\Rightarrow$: otherwise $(\rho,-[u]_{i^{\prime}})\in x^{i}$; $\Leftarrow$:
any $X_{(\mu,v)}$ in the above equation is $\sim_{i}$-saturated)
$\Leftrightarrow$ | $x\in X_{(\rho,[u]_{i^{\prime}})}$

Hence, for any $i\in\alpha$, $B^{X}$ is closed under taking
$\backsim_{i}$-saturation, and then is a cylindric basis about
$\\{\backsim_{i}:i\in\alpha\\}$.

  * •

$\\{D_{ij}:i,j\in\alpha\\}$ is diagonal family

For any $i\neq j\in\alpha$, let $x,y\in D_{ij}$ be different and $x\sim_{i}y$,
we know there is $(\rho,u)$ s.t. $x\in X_{(\rho,u)},y\in X_{(\rho,-u)}$, then
for $x_{\rho}:=\\{u:(\rho,u)\in x\\}$ and $y_{\rho}:=\\{u:(\rho,u)\in y\\}$,
we have $x_{\rho}\neq y_{\rho}$ and
$D^{\mathcal{C}}_{\rho^{-1}(i)\rho^{-1}(j)}\in x_{\rho}\cap y_{\rho}$. By
definition of atom, both $x_{\rho},y_{\rho}$ form ultrafilters on $B$ which is
the basis of $\mathcal{C}$, then by compactness and $T_{2}$ property, both
$\bigcap x_{\rho},\bigcap y_{\rho}$ are singletons. We denote points in
$\bigcap x_{\rho},\bigcap y_{\rho}$ as $a_{x},a_{y}$. By $x\sim_{i}y$, we know
$\\{u\in x_{\rho}:\rho^{-1}(i)\not\in\Delta(u)\\}=\\{u\in
y_{\rho}:\rho^{-1}(i)\not\in\Delta(u)\\}$, then
$a_{x}\sim_{\rho^{-1}(i)}a_{y}$ which contradicts
$D^{\mathcal{C}}_{\rho^{-1}(i)\rho^{-1}(j)}\in x_{\rho}\cap y_{\rho}$.

For the case of $\alpha=\beta$, let $f:|\mathcal{C}|\rightarrow
X_{\mathcal{C}}$ satisfy: for any
$a\in|\mathcal{C}|,f(a)\in\bigcap\\{X_{(1,u)}:u\mbox{ is clopen neighborhood
of }a\\}$ (It is easy to see that this set is a singleton. hence we actually
give a precise definition of $f$). Now we prove $f$ is a basis-preserving
C-mapping generally and is an injection if $\mathcal{C}_{\beta}$ is a $T_{2}$
space.

  * •

Continuity, basis-preserving, and diagonal keeping.

Here it is only proved that for the case of clopen sets in the cylindric
basis, the case of general open sets is an obvious corollary.

For any $X_{(\rho,u)}\in B^{X}$, we know there is $u^{\prime}\in B$,
$X_{(1,u^{\prime})}=X_{(\rho,u)}$. Now we prove, for any $u\in B$,
$f^{-1}[X_{(1,u)}]=u$:

By the definition of $f$, $f[u]\subseteq X_{(1,u)}$ i.e. $u\subseteq
f^{-1}[X_{(1,u)}]$. If there is $a\in f^{-1}[X_{(1,u)}]\backslash u$, we know
$a\in-u$, then $a\in f^{-1}[X_{(1,-u)}]=f^{-1}[-X_{(1,u)}]$, contradiction.

  * •

The case of $T_{2}$ space

For any $a,b\in|\mathcal{C}|$ s.t. $f(a)=f(b)$, we know
$\bigcap\\{X_{(1,u)}:u$ is a clopen neighborhood of
$a\\}=\bigcap\\{X_{(1,u)}:u$ is clopen neighborhood of $b\\}$, then
$\\{X_{(1,u)}:u$ is clopen neighborhood of $a\\}=\\{X_{(1,u)}:u$ is clopen
neighborhood of $b\\}$, then $\\{u:u$ is clopen neighborhood of $a\\}=\\{u:u$
is clopen neighborhood of $b\\}$, then $a=b$.

  * •

structure homomorphism and C-mapping

For any $i\in\beta$ and any $a\in|\mathcal{C}|$, if $b\sim_{i}a$, then
$\\{u:u$ is $i$-saturated clopen neighborhood of $a\\}=\\{u:u$ is
$i$-saturated clopen neighborhood of $b\\}$, then $\\{(1,u):u$ is
$i$-saturated clopen neighborhood of $a\\}=\\{(1,u):u$ is $i$-saturated clopen
neighborhood of $b\\}$, then $f(a)\sim_{i}f(b)$.

For any $X_{(1,u)}\in B^{X}$ s.t. $[f(a)]_{i}\cap X_{(1,u)}\neq\emptyset$, we
know $f(a)\in[X_{(1,u)}]_{i}=X_{(1,[u]_{i})}$, then $a\in[u]_{i}$, then there
is $b\sim_{i}a$ s.t. $b\in u$, and then $f(b)\in X_{(1,u)}$.

$\Box$

We call the $\mathcal{C}^{\alpha}$ in the above theorem the $\alpha$-expansion
space of $\mathcal{C}$.

###### Lemma 3.3

If $\mathcal{C}$ is an FOL space, then for any $x\in|\mathcal{C}^{\alpha}|$,
$\bigcap\\{u|(1,u)\in x\\}$ is a singleton.

Proof By the definition of atom, $\\{u|(1,u)\in x\\}$ has finite intersection
property, then by compactness, $\bigcap\\{u|(1,u)\in x\\}\neq\emptyset$.

If there are two different points $a,b\in\bigcap\\{u|(1,u)\in x\\}$, then
there are clopen set $v\in\tau^{\mathcal{C}}$ s.t. $a\in v,b\in-v$, then
neither $(1,v)$ nor $(1,-v)$ is in $x$, which contradicts the definition of
atom. Hence $\bigcap\\{u|(1,u)\in x\\}$ is singleton. $\Box$

###### Theorem 3.3

Let $\mathcal{C}_{\beta}$ be an FOL space, $\alpha\geq\beta$, then up to
S-homemorphism there is a unique $\alpha$-dimensional FOL space
$\mathcal{C^{\prime}}_{\alpha}$ and basis-preserving C-surjection $f$ s.t.
$f:\mathcal{C^{\prime}}_{\alpha}\rightarrow\mathcal{C}_{\beta}$.

Proof

  * •

Existence: Define $f:|\mathcal{C}^{\beta}|\rightarrow|\mathcal{C}|$ as
$x\mapsto\bigcap\\{u|(1,u)\in x\\}$, we verify that $f$ is a basis-preserving
C-surjection.

    * –

Continuity, basis-preserving and diagonal keeping

Here it is only proved that for the case of clopen sets in the cylindric
basis, the case of general open sets is an obvious corollary.

For any $X_{(\rho,u)}\in B^{X}$ s.t. $\Delta(X_{(\rho,u)})\subseteq\beta$, we
know $\rho[\Delta(u)]\subseteq\beta$, then there is $u^{\prime}\in B$ s.t.
$X_{(1,u^{\prime})}=X_{(\rho,u)}$ ($X_{1,u^{\prime}}=D_{ij}$ when
$u^{\prime}=D^{\mathcal{C}}_{ij}$). Now we just need to prove for any $u\in
B$, $f^{-1}[u]=X_{(1,u)}$:

$x\in f^{-1}[u]\Leftrightarrow f(x)\in u\Leftrightarrow(1,u)\in
x\Leftrightarrow x\in X_{(1,u)}$.

    * –

surjection

For any $a\in|\mathcal{C}|$, consider $x^{\prime}=\\{X_{(1,u)}:a\in u\\}$.
It’s easy to see $x^{\prime}$ can be expanded to an atom $x$, then by
definition of $f$, $f(x)=a$.

    * –

structure homomorphism and C-mapping

For any $x,y\in|\mathcal{C}^{\alpha}|$, and any $i\in\alpha$ s.t.
$x\sim_{i}y$, if $i\in\alpha\backslash\beta$ we know $f(x)=f(y)$; if
$i\in\beta$, then $\\{(1,u)\in x:i\not\in\Delta(u)\\}=\\{(1,u)\in
y:i\not\in\Delta(u)\\}$, then $\\{u\in B^{\mathcal{C}}:f(x)\in u\mbox{ and
}i\not\in\Delta(u)\\}=\\{u\in B^{\mathcal{C}}:f(y)\in u\mbox{ and
}i\not\in\Delta(u)\\}$, then by Proposition 2.5, $f(x)\sim_{i}f(y)$.

For any $u\in B,i\in\beta$,
$f^{-1}[[u]_{i}]=X_{(1,[u]_{i})}=[X_{(1,u)}]_{i}=[f^{-1}[u]]_{i}$.

  * •

Uniqueness:

Let $\mathcal{C}^{\prime}$ be a FOL space of $\alpha$ dimension,
$f^{\prime}:\mathcal{C}^{\prime}\rightarrow\mathcal{C}$ be a basis-preserving
C-surjection. denotes the cylindric basis of $\mathcal{C}^{\prime}$ as
$B^{\mathcal{C^{\prime}}}$, we know $\\{f^{\prime-1}[u]:u\in B\\}=\\{u\in
B^{\mathcal{C^{\prime}}}:\Delta(u)\subseteq\beta\\}$, then
$B^{\mathcal{C}^{\prime}}=\\{\rho(f^{\prime-1}[u]):u\in
B,\rho:\alpha^{\alpha}\\}=\\{\rho(f^{\prime-1}[u]):u\in
B,\rho:\alpha^{\beta}\mbox{ is injection }\\}$. For each
$a\in\mathcal{C^{\prime}}$, $a_{at}:=\\{(\rho,u)\in M\times B:a\in\rho
f^{\prime-1}[u]\\}$. By the definition of atom, $a_{at}$ is an atom in
$M\times B$ and from each atom in $M\times B$ we can get an ultrafilter on
$B^{\mathcal{C^{\prime}}}$, and then, by compactness, get a point $a$ in
$|\mathcal{C^{\prime}}|$ s.t. $a_{at}=x$. We define
$g:|\mathcal{C^{\prime}}|\rightarrow|\mathcal{C}^{\alpha}|$ as $a\mapsto
a_{at}$. Clearly, it is a bijection, and we have $f\circ g=f^{\prime}$.

Now, by Proposition 2.2, to prove $g$ is an S-homeomorphism, we just need to
prove that $g$ is a basis-preserving S-mapping:

    * –

Continuity, basis-preserving, and diagonal keeping

Here it is only proved that for the case of clopen sets in the cylindric
basis, the case of general open sets is an obvious corollary.

For $D_{ij}\in B^{X}$, we know there is $\rho\in
M,i^{\prime}j^{\prime}\in\alpha,D_{ij}=X_{(\rho,D^{\mathcal{C}}_{i^{\prime}j^{\prime}})}$
and $D^{\mathcal{C^{\prime}}}_{ij}=\rho
D^{\mathcal{C}}_{i^{\prime}j^{\prime}}$ (here
$\rho(i^{\prime})=i,\rho(j^{\prime})=j$).

Hence, we just need to prove, for any $X_{(\rho,u)}\in B^{X}$,
$g^{-1}[X_{(\rho,u)}]=\rho f^{\prime-1}[u]$:

$a\in\rho f^{\prime-1}[u]\Leftrightarrow(\rho,u)\in a_{at}=g(a)\Leftrightarrow
g(a)\in X_{(\rho,u)}$.

    * –

Structure homomophism:

For any $a,b\in|\mathcal{C^{\prime}}|$,

|  $i\in\alpha$, $a\sim_{i}b$
---|---
$\Rightarrow$ | there is no $u^{\prime}\in B^{\mathcal{C^{\prime}}}$ s.t. $a\in u^{\prime}\not\ni b$ and $i\not\in\Delta(u^{\prime})$
$\Rightarrow$ | there is no $u\in B$, $\rho:\alpha^{\alpha}$ s.t. $a\in\rho u\not\ni b$ and $i\not\in\rho[\Delta(u)]$
$\Rightarrow$ | there is no $u\in B$, $\rho:\alpha^{\alpha}$ s.t. $f(a)\in X_{(\rho,u)}\not\ni f(b)$ and $i\not\in\Delta(X_{(\rho,u)})$
$\Rightarrow$ | $f(a)\sim_{i}f(b)$

$\Box$

Call the mapping constructed in the above proof the $\alpha$-expansion mapping
of $\mathcal{C}_{\beta}$.

###### Remark 3.1

In fact, by constructing quotient spaces, we can easily prove the opposite
direction of this proposition, namely, for $\alpha\geq\beta$, each
$\alpha$-dimensional FOL space can be mapped by a basis-preserving
C-surjection to a $\beta$-dimensional FOL space, and this basis-preserving
C-surjection and this $\beta$-dimensional FOL space are unique up to
S-homeomorphism. This way, we get a one-to-one correspondence between FOL
spaces with different dimensions.

##  4 Topologization of first-order logic semantics

Based on the purely topological discussion in Section 3, this section
establishes a connection between first-order logic and cylindric space and
finally obtains a systematic topological representation of semantics of first-
order logic. The basic pattern of this connection has been shown by examples
in Section 2.2. To facilitate a rigorous discussion, in 4.3, we generalize the
correspondence between clopen sets in the topologization space or model space
and formulas in these examples as a mapping (called $\mathcal{L}$-formation)
from formulas of a language to clopen sets in a cylindric space. In 4.1, we
initially establish the connection between first-order structures and points
in FOL spaces. And on this basis, a topological proof of the first-order logic
compactness theorem is given. This means that all model spaces are FOL spaces.
In 4.4, for each infinite ordinal $\alpha$, each theory $T$ is associated with
a unique $\alpha$-dimensional FOL space called $\alpha$-model space of $T$;
and the class of $\alpha$-dimensional FOL spaces is shown to be the class of
S-homeomorphic spaces of $\alpha$-model spaces. In this way, we provide enough
points for representing uncountable models of $T$. In 4.1, the topological
representations of first-order structures and elementary embeddings are
finally given in two theorems, respectively.

###  4.1 $\alpha$-topologization and $\mathcal{L}$-formation

In subsection 2.2, we define the $\omega$-topologization of first-order
structure which can be easily generalized to any infinite ordinal $\alpha$:
for a first-order language $\mathcal{L}$ and an infinite ordinal $\alpha$, let
$\mathcal{L}_{\alpha}$ be the language obtained by expanding the set of
variables $\\{v_{i}:i\in\omega\\}$ in $\mathcal{L}$ to
$\\{v_{i}:i\in\alpha\\}$. The definition of $\mathcal{L}_{\alpha}$-formulas
and the definition of substitution for these formulas are then identical to
those in $\mathcal{L}$. For a $\mathcal{L}$-structure $\mathfrak{A}$, we
consider the $\alpha$ power of its domain $A$, written as $A^{\alpha}$, and
the clopen set $A_{\phi}$ corresponding to
$\phi\in\Lambda_{\mathcal{L}_{\alpha}}$. Based on this, applying exactly the
same method as that used to handle the $\omega$-topologization, we can obtain
the $\alpha$-topologization of $\mathfrak{A}$ (denoted as
$\mathcal{C}_{\alpha}^{\mathfrak{A}}$) and prove that
$\mathcal{C}_{\alpha}^{\mathfrak{A}}$ is also a basis-finite cylindric space.

From the construction of $\mathcal{C}_{\alpha}^{\mathfrak{A}}$, we naturally
obtain a mapping $\phi\mapsto A_{\phi}$ from $\Lambda_{\mathcal{L}_{\alpha}}$
to the cylindric basis of $\mathcal{C}_{\alpha}^{\mathfrak{A}}$, which can be
defined abstractly on a general basis-finite cylindric space as follows.

###### Definition 4.1

For an infinite ordinal $\alpha$, a basis-finite cylindric space
$\mathcal{C}_{\alpha}$, if the mapping
$l:\Lambda_{\mathcal{L}_{\alpha}}\rightarrow\tau_{\mathcal{C}}$ satisfies the
following conditions, then we call $l$ an $\mathcal{L}$-formation on
$\mathcal{C}_{\alpha}$:

  * •

For $n$-ary predicate $R$, $l(Rv_{0}...v_{n-1})$ is a clopen set s.t.
$\Delta(l(Rv_{0}...v_{n-1}))\subseteq n$;

  * •

For $i,j\in\alpha$, $l(v_{i}=v_{j})=D^{\mathcal{C}}_{ij}$;

  * •

For $i,j\in\alpha$, $l(\phi(v_{i}/v_{j}))=[l(\phi)\cap
D^{\mathcal{C}}_{ij}]_{j}$;

  * •

$l(\neg\phi)=|\mathcal{C}|\backslash l(\phi)$;

  * •

$l(\phi\wedge\psi)=l(\phi)\cap l(\psi)$;

  * •

$l(\exists v_{i}\phi)=[l(\phi)]_{i}$;

  * •

$ran(l)$ is a topological basis of $\tau_{\mathcal{C}}$.

For any $\mathcal{L}$-structure $\mathfrak{A}$ and its $\alpha$-topologization
$\mathcal{C}^{\mathfrak{A}}_{\alpha}$, define
$l_{\mathfrak{A}}:\Lambda_{\mathcal{L}_{\alpha}}\rightarrow\tau_{\mathcal{C}^{\mathfrak{A}}_{\alpha}}$
as $\phi\mapsto A_{[\phi]}$. It is easy to verify that $l_{\mathfrak{A}}$ is
an $\mathcal{L}$-formation. Call $l_{\mathfrak{A}}$ the natural
$\mathcal{L}$-formation on $\mathcal{C}^{\mathfrak{A}}_{\alpha}$.

For any $\mathcal{L}$-theory $T$ and its model space $\mathcal{C}^{T}$, let
$l_{T}:\Lambda_{\mathcal{L}}\rightarrow\tau_{\mathcal{C}^{T}}$ be $\phi\mapsto
S_{[\phi]}$. It is easy to verify that $l_{T}$ is also an
$\mathcal{L}$-formation. Call $l_{T}$ the natural $\mathcal{L}$-formation on
$\mathcal{C}^{T}$.

###### Definition 4.2

Let $f:\mathcal{C}_{\alpha}\rightarrow\mathcal{C^{\prime}}_{\beta}$ be an
S-mapping, $\mathcal{L}\geq\mathcal{L^{\prime}}$ be two languages,
$l,l^{\prime}$ be $\mathcal{L}$-formation on $\mathcal{C}$ and
$\mathcal{L^{\prime}}$-formation on $\mathcal{C^{\prime}}$ respectively, if
for any
$\phi\in\Lambda_{\mathcal{L^{\prime}}_{\beta}},f^{-1}[l^{\prime}(\phi)]=l(\phi)$,
then we say $f$ maps $l$ to $l^{\prime}$ and denote this as
$f:(\mathcal{C},l)\rightarrow(\mathcal{C}^{\prime},l^{\prime})$.

###### Proposition 4.1

Let $\mathcal{C}_{\alpha},\mathcal{C}^{\prime}_{\beta}$ be basis-finite
cylindric space, $\mathcal{L}$ be a language,
$f:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ be basis-preserving C-mapping,
then for any $\mathcal{L}$-formation $l^{\prime}$ on
$\mathcal{C}^{\prime}_{\beta}$, there is a unique $\mathcal{L}$-formation $l$
on $\mathcal{C}_{\alpha}$ s.t.
$f:(\mathcal{C},l)\rightarrow(\mathcal{C}^{\prime},l^{\prime})$.

Proof

  * •

Existence: Define $l:\Lambda_{\mathcal{L}_{\alpha}}\rightarrow
2^{\mathcal{C}}$ as: for any $n$-ary predicate $R$,
$l(Rv_{0}...v_{n-1}):=f^{-1}[l^{\prime}(Rv_{0}...v_{n-1})]$, for any other
formula $\phi$, $l(\phi)$ is determined by the inductive conditions in the
definition of $\mathcal{L}$-formation. By Proposition 2.3, it’s easy to see
that for any $\phi\in\Lambda_{\mathcal{L}_{\beta}}$,
$l(\phi)=f^{-1}[l^{\prime}(\phi)]$. Now we just need to prove $ran(l)$ forms a
basis of $\tau^{\mathcal{C}}$:

    * –

For any $n$-ary predicate $R$,
$l(Rv_{0}...v_{n-1})=f^{-1}[l^{\prime}(Rv_{0}...v_{n-1})]$ has finite
dimension, then by definition it is easy to see for any
$\phi\in\Lambda_{\mathcal{L}_{\alpha}}$, $l(\phi)$ is a clopen set with finite
dimension;

    * –

By the inductive conditions in the definition, $ran(l)$ is closed under taking
intersection, union, complement and $\sim_{i}$-saturation for any
$i\in\alpha$.

    * –

For any clopen set $u\in\tau^{\mathcal{C}}$ with finite dimension, we know
there is a $\rho\in\alpha^{\beta}$ and a finite-dimensional clopen set $v$
s.t. $\Delta(v)\subseteq\beta$ and $\rho v=u$, then there is a
$v^{\prime}\in\tau^{\mathcal{C^{\prime}}}$ s.t. $f^{-1}[v^{\prime}]=v$, by the
definition of $\mathcal{L}$-formation, there is a $U^{\prime}\subseteq
ran(l^{\prime})$ such that $v^{\prime}=\bigcup U^{\prime}$, then there is a
$U\subseteq ran(l)$ such that $v=\bigcup U$, then $u=\rho\bigcup
U=\bigcup\\{\rho w:w\in U\\}$. By the third clause of the inductive conditions
of definition of $\mathcal{L}$-formation, $\\{\rho w:w\in U\\}\subseteq
ran(l)$.

For any open set $u\in\tau^{\mathcal{C}}$, we know there is a family of finite
dimensional clopen sets $V\subseteq\tau^{\mathcal{C}}$ s.t. $u=\bigcup V$. As
proved above, for each $v\in V$, there is $U\subseteq ran(l)$, $v=\bigcup U$,
then there is a $U^{*}\subseteq ran(l)$ such that $u=\bigcup U^{*}$.

  * •

Uniqueness:  For any $\mathcal{L}$-formation $l^{*}$ on $\mathcal{C}$, since
the definition of $\mathcal{L}$-formation is inductive, if $l\neq l^{*}$, then
there must be predicate $R$ s.t. $l^{*}(Rv_{0}...v_{n-1})\neq
l(Rv_{0}...v_{n-1})=f^{-1}l^{\prime}(Rv_{0}...v_{n-1})$.

$\Box$

###### Proposition 4.2

Let $\mathcal{C}_{\alpha},\mathcal{C}^{\prime}_{\beta}$ be basis-finite
cylindric spaces, where $\mathcal{C}^{\prime}_{\beta}$ is a $T_{2}$ space,
$\mathcal{L}$ be a language, $l,l^{\prime}$ be $\mathcal{L}$-formations on
$\mathcal{C}_{\alpha},\mathcal{C}^{\prime}_{\beta}$ respectively. Then there
is at most one C-mapping
$f:(\mathcal{C},l)\rightarrow(\mathcal{C}^{\prime},l^{\prime})$.

Proof Assume there are
$f,f^{\prime}:(\mathcal{C},l)\rightarrow(\mathcal{C}^{\prime},l^{\prime})$
such that there is an $a\in|\mathcal{C}|$ satisfying $f(a)\neq f^{\prime}(a)$,
then since $ran(l^{\prime})$ is a topological basis and $\mathcal{C}$ is
$T_{2}$, we know there is a
$\phi\in\Lambda_{\mathcal{L}_{\beta}}\subseteq\Lambda_{\mathcal{L}_{\alpha}}$
s.t. $f(a)\in l^{\prime}(\phi)$, $f^{\prime}(a)\in l^{\prime}(\neg\phi)$, then
$a\in l(\phi)\cap l(\neg\phi)=\emptyset$, contradiction. $\Box$

###### Proposition 4.3

Let $\mathcal{C}_{\alpha},\mathcal{C}^{\prime}_{\beta}$ be basis-finite
cylindric spaces, $\mathcal{L},\mathcal{L^{\prime}}$ be two languages,
$l,l^{\prime}$ be $\mathcal{L}$-formation on $\mathcal{C}_{\alpha}$ and
$\mathcal{L^{\prime}}$-formation on $\mathcal{C}^{\prime}_{\beta}$
respectively. For a structure homomophism
$f:(|\mathcal{C}|,\sim_{i})_{i\in\alpha}\rightarrow(|\mathcal{C^{\prime}}|,\sim_{i})_{i\in\alpha}$
s.t. for any $\phi\in\Lambda_{\mathcal{L}_{\beta}}$,
$f^{-1}[l^{\prime}(\phi)]=l(\phi)$, we have
$f:(\mathcal{C},l)\rightarrow(\mathcal{C}^{\prime},l^{\prime})$ is a C-mapping
and if $\mathcal{L}=\mathcal{L^{\prime}}$, then $f$ is basis-preserving.

Proof Both $ran(l),ran(l^{\prime})$ are cylindric bases of corresponding
spaces, then $f$ is continuous and clearly preserves diagonal. Then by
Proposition 2.3.(2), the case of C-mapping clearly holds. If
$\mathcal{L}=\mathcal{L^{\prime}}$, then for each clopen set $l(\phi)$ in the
cylindric basis of $\mathcal{C}$ with dimension in $\beta$, we can assume the
$\phi$ is in $\Lambda_{\mathcal{L}_{\beta}}$ w.l.o.g., then
$f^{-1}[l^{\prime}(\phi)]=l(\phi)$. hence $f$ is basis-preserving. $\Box$

###  4.2 Compactness of model space

###### Definition 4.3

Let $\mathcal{C}_{\alpha}$ be an FOL space, $b\in|\mathcal{C}|$

  * •

If for any $i\in\alpha$ and any clopen set $u$, $b\in[u]_{i}\Rightarrow
b\in\bigcup_{j\in\alpha}u(\frac{j}{i})$, then we call $b$ a model point;

  * •

If $b$ is a model point and $\mathcal{D}^{\mathcal{C}}$ doesn’t cover the set
of all factors of $b$, then we call $b$ a big model point.

###### Lemma 4.1

Let $\mathcal{C}_{\alpha}$ be an FOL space, $b\in|\mathcal{C}|$ is a model
point, then for any $\mathcal{L}$-formation $l$ on $\mathcal{C}$, there is a
$\mathcal{L}$-structure $\mathfrak{A}$ and basis-preserving C-mapping
$f:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}_{\alpha},l)$
s.t. $b\in ran(f)$ and: 1), $\|\mathfrak{A}\|\leq\alpha$; 2), there is a
domain point in $f^{-1}[b]$; 3), for any $c\prec b$, $c\in ran(f)$; 4), for
any $c\in ran(f)$, if there is a domain point in $f^{-1}[c]$, then $c$ is a
model point.

Proof

Let $\alpha^{\prime}:=\\{i\in\alpha:\forall j<i,b\not\in
D^{\mathcal{C}}_{ij}\\}$.

  * •

We construct an $\mathcal{L}$-structure
$\mathfrak{A}=(A,R^{\mathfrak{A}}_{i})_{i\in I}$ as:
$A:=\\{x_{i}:i\in\alpha^{\prime}\\}$, we know $\|A\|\leq\alpha$; for an
$n$-ary predicate $R$ and $x_{i_{1}},...,x_{i_{n}}\in A$, $\langle
x_{i_{1}}...x_{i_{n}}\rangle\in R^{\mathfrak{A}}\Leftrightarrow b\in
l(Rv_{i_{1}}...v_{i_{n}})$.

  * •

For any sequence $a\in A^{\alpha}$, we know for any $i\in\alpha$, there is an
$i^{\prime}\in\alpha^{\prime}$ s.t. $a(i)=x_{i^{\prime}}$, let
$\rho_{a}:\alpha^{\alpha}$ be defined as $\rho_{a}(i)=i^{\prime}$. Define
mapping $f:A^{\alpha}\rightarrow|\mathcal{C}|$ as $a\mapsto\rho_{a}^{-1}b$.

  * •

We show $f$ maps $l_{\mathfrak{A}}$ to $l$:

Use induction on $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$:

BS: For any $n$-ary predicate $R$, $i_{1},...,i_{n}\in\alpha$,

| $a\in A_{Rv_{i_{1}}v_{i_{n}}}$
---|---
$\Leftrightarrow$ | $\langle a(i_{1})...a(i_{n})\rangle\in R^{\mathfrak{A}}$
$\Leftrightarrow$ | $\langle x_{\rho_{a}(i_{1})}...x_{\rho_{a}(i_{n})}\rangle\in R^{\mathfrak{A}}$
$\Leftrightarrow$ | $b\in l(Rv_{\rho_{a}(i_{1})}...v_{\rho_{a}(i_{n})})$
$\Leftrightarrow$ | $f(a)\in\rho_{a}^{-1}l(Rv_{\rho_{a}(i_{1})}...v_{\rho_{a}(i_{n})})=l(Rv_{i_{1}}...v_{i_{n}})$
$\Leftrightarrow$ | $a\in f^{-1}[l(Rv_{i_{1}}...v_{i_{n}})]$

Hence
$l_{\mathfrak{A}}(Rv_{i_{1}}...v_{i_{n}})=A_{Rv_{i_{1}}v_{i_{n}}}=f^{-1}[l(Rv_{i_{1}}...v_{i_{n}})]$.

IH: For $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$ with complexity less than $l$,
$l_{\mathfrak{A}}(\phi)=A_{\phi}=f^{-1}[l(\phi)]$.

IS: The complexity of $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$ is $l$:

The cases where $\phi$ has the form $\neg\psi,\psi\wedge\chi$ are trivial.
Here we consider the case $\phi$ has the form $\exists v_{i}\psi$:

| $a\in f^{-1}[l(\phi)]$
---|---
$\Leftrightarrow$ | For each $j\not\in\rho_{a}^{-1}[\rho_{a}(\Delta(l(\psi)))],b\in\rho_{a}l(\phi)=\rho_{a}[l(\psi)]_{i}=[\rho_{a}l(\psi(v_{j}/v_{i}))]_{\rho_{a}(j)}$
$\Leftrightarrow$ | there is a $j^{\prime}\in\alpha$, for each $j\not\in\rho_{a}^{-1}[\rho_{a}(\Delta(l(\psi)))]$, $b\in(\rho_{a}l(\psi(v_{j}/v_{i})))(\frac{j^{\prime}}{\rho_{a}(j)})$
|
$=\rho_{a}(j^{\prime}/j)(l(\psi)(\frac{j}{i}))=\rho_{a}(j^{\prime}/i)l(\psi)$666Here
$\rho(j^{\prime}/i)$ denotes such a mapping: its only difference from $\rho$
is that $\rho(j^{\prime}/i)$ shoots $i$ to $j^{\prime}$.(By the property of
model point)
$\Leftrightarrow$ | there is an $a^{\prime}$ s.t. $f(a^{\prime})\in l(\psi)$ and for some $j^{\prime}$, $\rho_{a^{\prime}}=\rho_{a}(j^{\prime}/i)$
| (The existence of $a^{\prime}$: Let $j^{\prime}$ be the least number
satisfying $b\in\rho_{a}(j^{\prime}/i)l(\psi)$, we can obtain $a^{\prime}$)
$\Leftrightarrow$ | there is an $a^{\prime}$ s.t. $a^{\prime}\sim_{i}a$ and $a^{\prime}\in A_{\psi}$
$\Leftrightarrow$ | $a\in A_{\phi}$

Hence, for any $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$,
$l_{\mathfrak{A}}(\phi)=A_{\phi}=f^{-1}[l(\phi)]$.

  * •

$f$ is a basis-preserving C-mapping

For any $i\in\alpha$,

| $a\sim_{i}a^{\prime}\in A^{\alpha}$
---|---
$\Rightarrow$ | for any $j\neq i,a(j)=a^{\prime}(j)$
$\Rightarrow$ | for any $j\neq i,\rho_{a}(j)=\rho_{a^{\prime}}(j)$
$\Rightarrow$ |  $\\{\rho_{a}^{-1}u:u$ is a clopen neighbourhood of $b$ and $\rho_{a}^{-1}u$ is $i$-saturated$\\}=$
|  $\\{\rho_{a^{\prime}}^{-1}u:u$ is a clopen neighbourhood of $b$ and
$\rho_{a}^{-1}u$ is $i$-saturated$\\}$
$\Rightarrow$ |  $\\{u:u$ is $i$-saturated clopen neighbourhood of $f(a)\\}=$
|  $\\{u:u$ is an $i$-saturated clopen neighbourhood of $f(a^{\prime})\\}$
$\Rightarrow$ | $f(a)\sim_{i}f(a^{\prime})$

As proven above, for any $\phi$,
$l_{\mathfrak{A}}(\phi)=A_{\phi}=f^{-1}[l(\phi)]$. Then by Proposition 4.3,
$f$ is a basis-preserving C-mapping.

  * •

$b\in ran(f)$ and there is a domain point in $f^{-1}[b]$

For any $i\in\alpha$, if $i\not\in\alpha^{\prime}$, then by definition of
$\alpha^{\prime}$ we know there is a $j<i$ s.t. $j\in\alpha^{\prime}$ and
$b\in D^{\mathcal{C}}_{ij}$. Write
$i^{\prime}:=argmin_{j}\\{x_{j}:j\in\alpha^{\prime},b\in
D^{\mathcal{C}}_{ij}\\}$, we know there is a sequence $a\in A^{\alpha}$ s.t.
$a(i)=i^{\prime}$, then it’s easy to see $f(a)=\rho_{a}^{-1}b=b$ and $a$ is a
domain point.

  * •

There is a $\rho:c\prec b\Rightarrow c\in ran(f)$

Let $b^{\prime}$ be a domain point in $f^{-1}[b]$, we define
$c^{\prime}\in|\mathcal{C}^{\mathfrak{A}}_{\alpha}|$ as
$c^{\prime}(i)=b^{\prime}(\rho(i))$. It’s easy to see for any $\phi$,
$c^{\prime}\in l_{\mathfrak{A}}(\phi)\Leftrightarrow b^{\prime}\in
l_{\mathfrak{A}}(\phi(\overline{v_{\rho(i)}}/\overline{v_{i}}))$, then
$f(c^{\prime})\in l(\phi)\Leftrightarrow b\in
l(\phi(\overline{v_{\rho(i)}}/\overline{v_{i}}))=\rho l(\phi)$, then
$f(c^{\prime})=\rho^{-1}b=c$.

  * •

There is a domain point in $f^{-1}[c]\Rightarrow c$ is a model point

Let $a$ be this domain point. It is easy to see for any $i\in\alpha$ and any
formula $\phi$,

| $a\in[l_{\mathfrak{A}}(\phi)]_{i}=l_{\mathfrak{A}}(\exists v_{i}\phi)$
---|---
$\Rightarrow$ | $a\in\bigcup_{j\in\alpha}\\{l_{\mathfrak{A}}(\phi(v_{j}/v_{i}))\\}=\bigcup_{j\in\alpha}l_{\mathfrak{A}}(\phi)(\frac{j}{i})$

hence if $c\in[l(\phi)]_{i}$, then
$c\in\bigcup_{j\in\alpha}l(\phi)(\frac{j}{i})$. hence $c$ is a model point.

$\Box$

###### Lemma 4.2

Let $\mathcal{C}_{\alpha}$ be an FOL space, $v$ be an $\emptyset$-dimensional
complete closed set on it s.t. the number of clopen sets in the subspace
induced by $v$ is not greater than $\alpha$, then for any $a\in v$, there is a
model point $b\in v$ s.t. $a\prec b$. In particular, such a big model point
exists if the diagonal family has no finite subset covering $v$.

Proof Let $\lambda:\omega\times\alpha\rightarrow\alpha$ be a bijection,
$\rho:\alpha\rightarrow\alpha$ be an injection s.t.
$ran(\rho)=\lambda[1\times\alpha]$, then $\rho\\{a\\}$ is a non-empty complete
closed set of dimension $\lambda[1\times\alpha]$. Use induction on
$n\in\omega$, we construct closed sets $X_{n}$ and families $U_{n}$ of clopen
sets as follow:

  * •

BS:  $X_{0}:=\rho\\{a\\}$, $U_{0}:=\\{[u]_{i}:u\mbox{ is clopen
set},\rho\\{a\\}\subseteq[u]_{i}\mbox{ but for any
}j\in\lambda[1\times\alpha],\rho\\{a\\}\not\subseteq u(\frac{j}{i})\\}$.

  * •

IH:  For $m<n$, $X_{m}$ is complete closed set of dimension
$\lambda[(m+1)\times\alpha]$ s.t. $X_{m}\subseteq\rho\\{a\\}$;
$U_{m}:=\\{[u]_{i}:u\mbox{ is clopen
set},i\in\alpha,X_{m}\subseteq[u]_{i}\backslash\bigcup_{j\in\lambda[(m+1)\times\alpha]}u(\frac{j}{i})\\}$.

  * •

IS:  It is easy to see $\|U_{n-1}\|\leq\alpha$. Lists $U_{n-1}$ as
$\\{[u_{j}]_{i_{j}}:j\in\alpha,i_{j}\in\alpha\\}$. For $j\in\alpha$, let
$j^{\prime}:=\lambda(n,j)$, $u^{\prime}_{j}:=u_{j}(\frac{j^{\prime}}{i_{j}})$.
Then for any $j_{1},...,j_{k}\in\alpha$,

| $a\in X_{n-1}$
---|---
$\Leftrightarrow$ | $a\in X_{n-1}\cap[u_{j_{1}}]_{i_{j_{1}}}$
$\Leftrightarrow$ |  $a\in X_{n-1}$ and there is $b\sim_{j_{1}^{\prime}}a$ s.t. $b\in u_{j_{1}}(\frac{j_{1}^{\prime}}{i_{j_{1}}})$
$\Leftrightarrow$ |  $a\in X_{n-1}$ and there is $b\sim_{j_{1}^{\prime}}a$ s.t. $b\in X_{n-1}\cap u_{j_{1}}(\frac{j_{1}^{\prime}}{i_{j_{1}}})$
$\Leftrightarrow$ | there is $b\sim_{j_{1}^{\prime}}a$ s.t. $b\in X_{n-1}\cap u_{j_{1}}(\frac{j_{1}^{\prime}}{i_{j_{1}}})$
$\Leftrightarrow$ | $a\in[X_{n-1}\cap u^{\prime}_{j_{1}}]_{j_{1}^{\prime}}$
| $\dots\dots$
$\Leftrightarrow$ | $a\in[...[X_{n-1}\cap u^{\prime}_{j_{1}}\cap...\cap u^{\prime}_{j_{k}}]_{j_{1}^{\prime}}...]_{j_{k}^{\prime}}$

Since $X_{n-1}\neq\emptyset$, we know $[...[X_{n-1}\cap
u^{\prime}_{j_{1}}\cap...\cap
u^{\prime}_{j_{k}}]_{j_{1}^{\prime}}...]_{j_{k}^{\prime}}\neq\emptyset$, then
$X_{n-1}\cap u^{\prime}_{j_{1}}\cap...\cap u^{\prime}_{j_{k}}\neq\emptyset$.
By compactness,
$X^{\prime}_{n}:=X_{n-1}\cap\bigcap_{j\in\alpha}u_{j}(\frac{j^{\prime}}{i_{j}})\neq\emptyset$
and easy to see $\Delta(X^{\prime}_{n})\subseteq\lambda[(n+1)\times\alpha]$.
Let $X_{n}$ be complete closed set in $X^{\prime}_{n}$ of dimension
$\lambda[(n+1)\times\alpha]$. $U_{n}:=\\{[u]_{i}:u\mbox{ is clopen
set},i\in\alpha,X_{n}\subseteq[u]_{i}\backslash\bigcup_{j\in\lambda[(n+1)\times\alpha]}u(\frac{j}{i})\\}$.

Let $X:=\bigcap_{n\in\omega}X_{n}$, then by compactness and $T_{2}$ property,
we know $X$ is a singleton. Let the point in $X$ be $b$, we know for any
$[u]_{j}\ni b$, there is a $j\in\alpha$ s.t. $u(\frac{i}{j})\ni b$, Hence $b$
is a model point. Since $b\in X\subseteq\rho\\{a\\}$, we know $a\prec b$. Then
the $\emptyset$-dimensional complete closed set containing $b$ is the same as
the $\emptyset$-dimensional complete closed set containing $a$. Hence $b\in
v$.

If the diagonal family has no finite subset covering $v$, then by compactness,
$v\cap\bigcap\\{-D_{ij}:i,j\in\alpha\\}\neq\emptyset$. Choosing an arbitrary
point $b$ in it, let $\rho,\rho^{\prime}$ be defined as
$\rho(i)=2i,\rho^{\prime}(i)=2i+1$, it is easy to see
$\rho\\{a\\}\cap\rho^{\prime}\\{b\\}\neq\emptyset$. For any point $a^{\prime}$
in it, let $a^{\prime}$ replace $a$ in the BS of induction. Obviously, the
last constructed model point takes $b$ as a factor and therefore is a big
model point. $\Box$

The following theorem implies the compactness theorem of first-order logic.

######  Theorem 4.1 (Compactness theorem of first-order logic)

For any first-order language $\mathcal{L}$ and any $\mathcal{L}$-theory $T$,
$\mathcal{C}^{T}$ is compact and therefore is an FOL space.

Proof By Theorem 3.2, there is a basis-preserving C-injection
$f:\mathcal{C}^{T}\rightarrow(\mathcal{C}^{T})^{\omega}$. We know
$(\mathcal{C}^{T})^{\omega}$ is an FOL space. Now we just need to prove $f$ is
a surjection which, by Lemma 2.2, means $\mathcal{C}^{T}$ is S-homemorphic to
$(\mathcal{C}^{T})^{\omega}$.

Let $l:\Lambda_{\mathcal{L}}\rightarrow\tau^{X}$777$\tau^{X}$ is the topology
of $(\mathcal{C}^{T})^{\omega}$. be $\phi\mapsto X_{(1,l_{T}(\phi))}$, we know
$f^{-1}[l(\phi)]=l_{T}(\phi)$. From the structure of $\omega$-expansion space,
it is easy to see that $l$ is an $\mathcal{L}$-formation. Hence
$f:(\mathcal{C}^{T},l_{T})\rightarrow((\mathcal{C}^{T})^{\omega},l)$.

For any $x\in(\mathcal{C}^{T})^{\omega}$, let the number of clopen sets
containing $x$ be $\alpha$, we know $\alpha\geq\omega$. Denote
$\mathcal{C}^{\prime}:=((\mathcal{C}^{T})^{\omega})^{\alpha}$,
$g:\mathcal{C}^{\prime}\rightarrow(\mathcal{C}^{T})^{\omega}$ be expansion
mapping. By Proposition 4.1, there is a unique $\mathcal{L}$-formation
$l^{\prime}$ on $\mathcal{C}^{\prime}$ s.t.
$g:(\mathcal{C}^{\prime},l^{\prime})\rightarrow((\mathcal{C}^{T})^{\omega},l)$.

Choosing an arbitrary $z\in g^{-1}[x]$, the number of clopen sets in
$\mathcal{C}^{\prime}$ containing $z$ is $\alpha\times\alpha=\alpha$, then by
Lemma 4.2, there is a $\rho:\alpha^{\alpha}$ and a model point $y\in
z|_{\emptyset}$ s.t. $\rho^{-1}y=z$. By Lemma 4.1, there is a
$\mathcal{L}$-structure $\mathfrak{A}$ and a basis-preserving C-mapping
$h:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{\prime},l^{\prime})$
s.t. $\|\mathfrak{A}\|\leq\alpha$ and $z\in ran(h)$ i.e. there is an $a\in
A^{\alpha}$, $h(a)=z$.

Clearly $a\in\bigcap\\{A_{\phi}:x\in l(\phi)\\}$, which means that
$\\{\phi\in\Lambda_{\mathcal{L}}:x\in l(\phi)\\}$ is satisfied by the
assignment represented by $a$. Then there exists a point in
$\bigcap\\{S_{\phi}:x\in l(\phi)\\}$ which is mapped to $x$. By the
arbitrariness of $x$, we know that $f$ is a surjection. $\Box$

###  4.3 $\alpha$-model space

Theorem 4.1 tells us that the model space of a first-order logic theory must
be an FOL space. In turn, for any FOL space $\mathcal{C}$, it is easy to see
that we can take a formation on it (e.g., take a language $\mathcal{L}$ such
that for any $n$, $\mathcal{L}$ contains $|\tau^{\mathcal{C}}|$ $n$-ary
predicates, and then construct the formation such that each clopen set $u$ in
$\mathcal{C}$ with $\Delta(u)=n$ is mapped by an $n$-ary atomic formula) and
thus determine a first-order logic theory according to the following theorem.

###### Theorem 4.2

Let $\mathcal{C}_{\alpha}$ be an FOL space, $\mathcal{L}$ be first-order
language, $l$ be a $\mathcal{L}$-formation on $\mathcal{C}_{\alpha}$,
$T:=\\{\phi\in\Lambda_{\mathcal{L}}:\phi\mbox{ is a
sentence},l(\phi)=|\mathcal{C}|\\}$, then $T$ is an $\mathcal{L}$-theory and
there is an S-homemorphism
$f:(\mathcal{C},l)\cong((\mathcal{C}^{T})^{\alpha},l^{\alpha}_{T})$ (here
$l^{\alpha}_{T}$ is the $\mathcal{L}$-formation on
$(\mathcal{C}^{T})^{\alpha}$ induced by $l_{T}$).

Proof

  * •

$T$ is a theory

It suffices to show that $T$ is closed under consequence.

For any sentence $\phi\in\Lambda_{\mathcal{L}}$ s.t. $T\vDash\phi$, let
$b\in|\mathcal{C}|$ be an arbitrary model point. By Lemma 4.1, there is an
$\mathcal{L}$-structure $\mathfrak{A}$, a basis-preserving C-mapping
$f:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C},l)$
and a domain point $b^{\prime}\in f^{-1}[b]$, then
$b^{\prime}\in\bigcap_{\psi\in T}l_{\mathfrak{A}}(\psi)$ i.e. the assignment
represented by $b^{\prime}$ realizes $T$, then the assignment represented by
$b^{\prime}$ satisfies $\phi$, then $b\in l(\phi)$. Since $\phi$ is a
sentence, we know the dimension of $l(\phi)$ is $\emptyset$, then for any
$\rho:\alpha^{\alpha}$, $\rho l(\phi)=l(\phi)$, then the factor set of $b$ is
subset of $l(\phi)$. By Lemma 4.2, $|\mathcal{C}|=\\{a\in|\mathcal{C}|:\mbox{
there is a model point }b\mbox{ }s.t.\mbox{ }a\prec b\\}$, then from the
arbitrariness of $b$ in the above proof, we know $|\mathcal{C}|\subseteq
l(\phi)$ and therefore $\phi\in T$.

  * •

Constructing f as a bijection

For any $\phi,\psi\in\Lambda_{\mathcal{L}_{\alpha}}$,

| $l(\phi)\cap l(\psi)\neq\emptyset$
---|---
$\Leftrightarrow$ | $l(\phi\wedge\psi)\neq\emptyset$
$\Leftrightarrow$ |  $[...[l(\phi\wedge\psi)]_{i_{1}}...]_{i_{k}}\neq\emptyset$ (here $\\{i_{1},...,i_{k}\\}=\Delta(l(\phi\wedge\psi))$)
$\Leftrightarrow$ | $-[...[-l(\neg(\phi\wedge\psi))]_{i_{1}}...]_{i_{k}}\neq|\mathcal{C}|$
$\Leftrightarrow$ | $l(\forall v_{i_{1}}...v_{i_{k}}\neg(\phi\wedge\psi))\neq|\mathcal{C}|$
$\Leftrightarrow$ | $l(\forall v_{1}...v_{k}\neg(\phi\wedge\psi)(v_{1}/v_{i_{1}})...(v_{k}/v_{i_{k}}))\neq|\mathcal{C}|$
$\Leftrightarrow$ | $\forall v_{1}...v_{k}\neg(\phi\wedge\psi)(v_{1}/v_{i_{1}})...(v_{k}/v_{i_{k}})\not\in T$
$\Leftrightarrow$ | $l_{T}^{\alpha}(\forall v_{1}...v_{k}\neg(\phi\wedge\psi)(v_{1}/v_{i_{1}})...(v_{k}/v_{i_{k}}))\neq|(\mathcal{C}^{T})^{\alpha}|$
| $\dots\dots$
$\Leftrightarrow$ | $l_{T}^{\alpha}(\phi)\cap l_{T}^{\alpha}(\psi)\neq\emptyset$

Hence for any finite $\Phi\subseteq\Lambda_{\mathcal{L}_{\alpha}}$,
$\bigcap_{\phi\in\Phi}l(\phi)\neq\emptyset\Leftrightarrow\bigcap_{\phi\in\Phi}l_{T}^{\alpha}(\phi)\neq\emptyset$,
then by compactness, for any $\Phi\subseteq\Lambda_{\mathcal{L}_{\alpha}}$,
$\bigcap_{\phi\in\Phi}l(\phi)\neq\emptyset\Leftrightarrow\bigcap_{\phi\in\Phi}l_{T}^{\alpha}(\phi)\neq\emptyset$.
Hence, defining $f$ as $f(a):=a^{\prime}\in\bigcap\\{l_{T}^{\alpha}(\phi):a\in
l(\phi)\\}$ (by $T_{2}$ property, $a^{\prime}$ is unique), it is easy to see
$f$ is a bijection.

  * •

$f$ is S-homemophism

By definition of $f$, it is easy to see for any $\phi$,
$l(\phi)=f^{-1}[l_{\mathfrak{A}}^{\alpha}(\phi)]$, then for any $i\in\alpha$,
$a\sim_{i}b\in|\mathcal{C}|$, we know $\\{u\in\tau^{\mathcal{C}}:u$ is an
$i$-saturated clopen neighborhood of $a\\}=\\{u\in\tau^{\mathcal{C}}:u$ is an
$i$-saturated clopen neighborhood of $b\\}$, then
$\\{\phi\in\Lambda_{\mathcal{L}_{\alpha}}:a\in l(\phi)$ and $v_{i}$ does not
occur freely in $\phi\\}=\\{\phi\in\Lambda_{\mathcal{L}_{\alpha}}:b\in
l(\phi)$ and $v_{i}$ does not occur freely in $\phi\\}$, then
$\\{\phi\in\Lambda_{\mathcal{L}_{\alpha}}:f(a)\in
l_{\mathfrak{A}}^{\alpha}(\phi)$ and $v_{i}$ does not occur freely in
$\phi\\}=\\{\phi\in\Lambda_{\mathcal{L}_{\alpha}}:f(b)\in
l_{\mathfrak{A}}^{\alpha}(\phi)$ and $v_{i}$ does not occur freely in
$\phi\\}$, then $\\{u:u$ is a $i$-saturated clopen neighborhood of
$f(a)\\}=\\{u:u$ is a $i$-saturated clopen neighborhood of $f(b)\\}$, then by
Proposition 2.5, $f(a)\sim_{i}f(b)$, then by Proposition 4.3, $f$ is a basis-
preserving C-mapping, then by Lemma 2.2, $f$ is an S-homemorphism.

$\Box$
This theorem implies that the class of all FOL spaces is exactly the class of
all S-homeomorphic spaces of model spaces and their expansion spaces. Based on
the discussion above, we can have a more general definition of model space.

###### Definition 4.4

Let $\mathcal{C}_{\alpha}$ be an FOL space, $\mathcal{L}$ be a first-order
language, $l$ be an $\mathcal{L}$-formation on $\mathcal{C}_{\alpha}$, $T$ be
an $\mathcal{L}$-theory, if
$T=\\{\phi\in\Lambda_{\mathcal{L}}:l(\phi)=|\mathcal{C}|\\}$, then we call
$\mathcal{C}_{\alpha}$ an $\alpha$-dimensional model space of $T$ and denote
it as $\mathcal{C}^{T}_{\alpha}$, call $l$ a $T$-formation on it and denote it
as $l^{\alpha}_{T}$.

###  4.4 The Representation of Semantics of First-Order Logic

Lemma 4.1 tells us that a model of a theory $T$ of cardinality $\kappa$ can be
represented as a cylindric space that can be mapped to the $\kappa$-model
space of $T$ by a basis-preserving C-mapping and therefore as a model point in
this FOL space. We show this correspondence between model points and models in
more detail in the first theorem of this section. Based on this, we will see
that an elementary embedding between first-order models can be represented as
a basis-preserving C-mapping between the corresponding topologization spaces
and, therefore, as a factoring relationship between two model points in a FOL
space. In this way, a systematic topological representation is presented.

###### Theorem 4.3

Let $\mathcal{L}$ be a first-order language, $T$ be an $\mathcal{L}$-theory,

  1. 1.

For any $T$-model $\mathfrak{A}$, any infinite ordinal $\alpha\geq\beta$,
there exists a unique basis-preserving C-mapping
$f:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{T}_{\beta},l_{T}^{\beta})$;

  2. 2.

For any $T$-model $\mathfrak{A}$, any infinite ordinal $\alpha$ and basis-
preserving C-mapping
$f:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha})$,
let $U$ be the non-empty set of all domain points in
$\mathcal{C}^{\mathfrak{A}}_{\alpha}$, then $f[U]$ is an $\asymp$-equivalence
class and all the points in it are model point and if
$\alpha=\|\mathfrak{A}\|$, then these model points are big.

  3. 3.

For any infinite ordinal $\alpha$, model point
$a\in|\mathcal{C}^{T}_{\alpha}|$, up to S-homeomorphism, there exists unique
$T$-model $\mathfrak{A}$ s.t. for the basis-preserving C-mapping
$f:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha})$,
there is domain point $b\in f^{-1}[a]$. The cardinality of $|\mathfrak{A}|$ is
not greater than $\alpha$. We say $a$ represents $\mathfrak{A}$.

Proof

  1. 1.

     * •

Existence: For any $a\in|\mathcal{C}^{\mathfrak{A}}_{\alpha}|$, we know
$\\{l_{\mathfrak{A}}(\phi):\phi\in\Lambda_{\mathcal{L}_{\alpha\cap\beta}},a\in
l_{\mathfrak{A}}(\phi)\\}$ is closed under taking finite intersection and does
not contain $\emptyset$, then it is easy to see
$X:=\\{l_{T}(\phi):\phi\in\Lambda_{\mathcal{L}_{\alpha\cap\beta}},a\in
l_{\mathfrak{A}}(\phi)\\}$ has the same property, then by compactness,
$\bigcap X\neq\emptyset$. Let $a^{\prime}\in\bigcap
X\cap\bigcap_{i\in\beta\backslash\alpha}D^{T}_{0i}$ (by $T_{2}$ property and
property of diagonal family, this is a singleton), we define
$f:|\mathcal{C}^{\mathfrak{A}}_{\alpha}|\rightarrow|\mathcal{C}^{T}_{\beta}|$
as $f(a)=a^{\prime}$, then for any
$\psi\in\Lambda_{\mathcal{L}_{\alpha\cap\beta}}$,
$b\in|\mathcal{C}^{\mathfrak{A}}_{\alpha}|$, $b\in
l_{\mathfrak{A}}(\psi)\Leftrightarrow\psi\in\\{\phi\in\Lambda_{\mathcal{L}_{\alpha\cap\beta}}:b\in
l_{\mathfrak{A}}(\phi)\\}\Leftrightarrow f(b)\in
l_{T}^{\beta}(\psi)\Leftrightarrow b\in f^{-1}[l_{T}^{\beta}(\psi)]$, then
$l_{\mathfrak{A}}(\psi)=f^{-1}[l_{T}^{\beta}(\psi)]$.

As with the mapping in the proof of Lemma 4.1, it is easy to verify that $f$
is a structural homomorphism. Then by Proposition 4.3,
$f:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{T}_{\beta},l_{T}^{\beta})$
is a basis-preserving C-mapping.

     * •

Uniqueness: The uniqueness can be obtained immediately by Proposition 4.2.

  2. 2.

For $a,b\in|\mathcal{C}^{\mathfrak{A}}_{\alpha}|$ s.t. $a$ is a domain point,
define $\rho:\alpha^{\alpha}$ as $\rho(i):=min\\{j:b(i)=a(j)\\}$, we know for
any $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$, $b\in
l_{\mathfrak{A}}(\phi)\Leftrightarrow a\in\rho
l_{\mathfrak{A}}(\phi)=l_{\mathfrak{A}}((\overline{v_{\rho(i)}}/\overline{v_{i}}))$,
then

|  $b$ is a domain point
---|---
$\Leftrightarrow$ |  $b$ is a domain point and $a\in\bigcap\\{l_{\mathfrak{A}}(\phi(\overline{v_{\rho(i)}}/\overline{v_{i}})):b\in l_{\mathfrak{A}}(\phi)\\}$
$\Leftrightarrow$ |  $a\in\bigcap\\{l_{\mathfrak{A}}(\phi(\overline{v_{\rho(i)}}/\overline{v_{i}})):b\in l_{\mathfrak{A}}(\phi)\\}$ and for any $i\not\in ran(\rho)$,
| there is an $i^{\prime}\in ran(\rho)$ s.t. $a\in
l_{\mathfrak{A}}(v_{i}=v_{i^{\prime}})$
$\Leftrightarrow$ |  $f(a)\in\bigcap\\{l_{T}^{\alpha}(\phi(\overline{v_{\rho(i)}}/\overline{v_{i}})):f(b)\in l_{T}^{\alpha}(\phi)\\}$ and for any $i\not\in ran(\rho)$,
| there is an $i^{\prime}\in ran(\rho)$ s.t. $f(a)\in
l_{T}^{\alpha}(v_{i}=v_{i^{\prime}})$
$\Leftrightarrow$ |  $f(a)\in\bigcap\\{\rho(l_{T}^{\alpha}(\phi)):f(b)\in l_{T}^{\alpha}(\phi)\\}$ and for any $i\not\in ran(\rho)$,
| there is an $i^{\prime}\in ran(\rho)$ s.t. $f(a)\in
l_{T}^{\alpha}(v_{i}=v_{i^{\prime}})$
$\Leftrightarrow$ |  $\rho:f(b)\prec f(a)$ and for any $i\not\in ran(\rho)$,
| there is an $i^{\prime}\in ran(\rho)$ s.t. $f(a)\in D^{T}_{ii^{\prime}}$
($ran(l_{T}^{\alpha})$ is cylindrical basis)
$\Leftrightarrow$ | $f(b)\asymp f(a)$

Hence, the image of the non-empty set of all domain points under $f$ is an
$\asymp$-equivalence class. By Lemma 4.1, all points in this class are model
points.

If $\alpha=\|\mathfrak{A}\|$, then there is a domain point $a\in A^{\alpha}$
s.t. for any $i\neq j,a(i)\neq a(j)$, then
$f(a)\not\in\bigcup\mathcal{D}^{T}$, then $f(a)$ is a big model point. For
other $b\in U$, we know $f(a)\prec b$ and therefore $b$ is also big.

  3. 3.

Existence is exactly Lemma 4.1. To prove uniqueness, consider two basis-
preserving C-mapping
$f:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha})$
and
$f:(\mathcal{C}^{\mathfrak{A}^{\prime}}_{\alpha},l_{\mathfrak{A}^{\prime}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha})$
s.t. there are domain points $b\in f^{-1}[a],b^{\prime}\in f^{\prime-1}[a]$.
for any $\rho:\alpha^{\alpha}$, any $c\in A^{\alpha}$, we define $\rho^{+}c\in
A^{\alpha}$ as $\rho^{+}c(i)=c(\rho(i))$. it is easy to see
$A^{\alpha}=\\{\rho^{+}b:\rho:\alpha^{\alpha}\\}$ and similarly,
$A^{\prime\alpha}=\\{\rho^{+}b^{\prime}:\rho:\alpha^{\alpha}\\}$. Define
$h:A\rightarrow A^{\prime}$ as $\rho^{+}b\mapsto\rho^{+}b^{\prime}$, we show
$h:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{\mathfrak{A}^{\prime}}_{\alpha},l_{\mathfrak{A}^{\prime}})$
is S-homeomorphism:

     * •

$h$ is well-defined and is an injection:

For any $\rho,\rho^{\prime}:\alpha^{\alpha}$,

| $\rho^{+}b=\rho^{\prime+}b$
---|---
$\Leftrightarrow$ | For any $i$, $b(\rho(i))=b(\rho^{\prime}(i))$
$\Leftrightarrow$ | For any $i$, $b\in l_{\mathfrak{A}}(v_{\rho(i)}=v_{\rho^{\prime}(i)})$
$\Leftrightarrow$ | For any $i$, $b^{\prime}\in l_{\mathfrak{A^{\prime}}}(v_{\rho(i)}=v_{\rho^{\prime}(i)})$
| $\dots\dots$
$\Leftrightarrow$ | $h(\rho^{+}b)=\rho^{+}b^{\prime}=\rho^{\prime+}b^{\prime}=h(\rho^{\prime+}b^{\prime})$

     * •

$h$ is a surjection:
$A^{\prime\alpha}=\\{\rho^{+}b^{\prime}:\rho:\alpha^{\alpha}\\}=ran(h)$.

     * •

$h$ is a basis-preserving C-mapping:

For any $i\in\alpha$ and any $a,c\in A^{\alpha}$ s.t. $a\sim_{i}c$, we know
$a(j)=b(j)$ for any $j\neq i$, then there are
$\rho,\rho^{\prime}:\alpha^{\alpha}$ s.t. $\rho(j)=\rho^{\prime}(j)$ for any
$j\neq i$ and $a=\rho^{+}b,c=\rho^{\prime+}b$, then
$h(a)=\rho^{+}b^{\prime},h(c)=\rho^{\prime+}b^{\prime}$, then
$h(a)(j)=h(c)(j)$ for any $j\neq i$, then $h(a)\sim_{i}h(c)$.

For any $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$,

| $\rho^{+}b\in l_{\mathfrak{A}}(\phi)$
---|---
$\Leftrightarrow$ |  $b\in l_{\mathfrak{A}}(\phi)(\frac{\rho(i_{1})}{i_{1}})...(\frac{\rho(i_{k})}{i_{k}})$ ($\Delta(l(\phi))=\\{i_{1},...,i_{k}\\}$)
$\Leftrightarrow$ | $b\in l_{\mathfrak{A}}(\phi(v_{\rho(i_{1})}/v_{i_{1}})...(v_{\rho(i_{k})}/v_{i_{k}}))$
$\Leftrightarrow$ | $b^{\prime}\in l_{\mathfrak{A^{\prime}}}(\phi(v_{\rho(i_{1})}/v_{i_{1}})...(v_{\rho(i_{k})}/v_{i_{k}}))$
| $\dots\dots$
$\Leftrightarrow$ | $\rho^{+}b^{\prime}\in l_{\mathfrak{A^{\prime}}}(\phi)$

By Proposition 4.3, $h$ is a basis-preserving C-mapping.

By Proposition 2.2, $h$ is an S-homeomorphism. By Proposition 2.4,
$f^{\prime}\circ h$ is a basis-preserving C-mapping and obviously maps
$l_{\mathfrak{A}}$ to $l_{T}^{\alpha}$, then by Proposition 4.2,
$f^{\prime}\circ h=f$.

$\Box$

###### Theorem 4.4

Let $\mathcal{L}$ be a first-order language, $\mathfrak{A},\mathfrak{B}$ be
$\mathcal{L}$-structures, the following statements are equivalent:

  1. 1.

There is an elementary embedding $f:\mathfrak{A}\rightarrow\mathfrak{B}$;

  2. 2.

For any infinite ordinal $\alpha$, there is a basis-preserving C-mapping
$h:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{\mathfrak{B}}_{\alpha},l_{\mathfrak{B}})$;

  3. 3.

For $T\subseteq Th(\mathfrak{A})\cap Th(\mathfrak{B})$, infinite ordinal
$\alpha\geq\|\mathfrak{A}\|\cup\|\mathfrak{B}\|$ and model points
$a,b\in|\mathcal{C}^{T}_{\alpha}|$ s.t. $a$ represents $\mathfrak{A}$, $b$
represents $\mathfrak{B}$, we have $a\prec b$.

Proof

  * •

$1\Rightarrow 2:$ Define $h:A^{\alpha}\rightarrow B^{\alpha}$ as
$h(a)(i):=f(a(i))$, then clearly the condition of structure homomorphism
holds. For any $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$, $c\in A^{\alpha}$,

| $c\in l_{\mathfrak{A}}(\phi)$
---|---
$\Leftrightarrow$ |  $\langle c(i_{1}),...,c(i_{k})\rangle\in\phi(\mathfrak{A})$ ($v_{i_{1}},...,v_{i_{k}}$ are the free variables in $\phi$)
$\Leftrightarrow$ | $\langle f(c(i_{1})),...,f(c(i_{k}))\rangle\in\phi(\mathfrak{B})$
$\Leftrightarrow$ | $h(c)\in l_{\mathfrak{B}}(\phi)$
$\Leftrightarrow$ | $c\in h^{-1}[l_{\mathfrak{B}}(\phi)]$

By Proposition 4.3, $h$ is a basis-preserving mapping.

  * •

$2\Rightarrow 3:$ By Theorem 4.3.(1), there are unique basis-preserving
C-mappings
$g:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha}),g^{\prime}:(\mathcal{C}^{\mathfrak{B}}_{\alpha},l_{\mathfrak{B}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha})$.
Since $\alpha\geq\|\mathfrak{A}\|\cup\|\mathfrak{B}\|$, we know both
$A^{\alpha}$ and $B^{\alpha}$ contain domain points. By Theorem 4.3.(2), there
are model points $a,b\in|\mathcal{C}^{T}_{\alpha}|$ representing
$\mathfrak{A},\mathfrak{B}$ respectively.

For any domain points $a^{*}\in g^{-1}[a]$,, $b^{*}\in g^{\prime-1}[b]$ and
any $i\in\alpha$, we know there is an $i^{\prime}$ s.t.
$b^{*}(i^{\prime})=h(a^{*})(i)$. Define $\rho:\alpha^{\alpha}$ as $i\mapsto
min\\{i^{\prime}:b^{*}(i^{\prime})=h(a^{*})(i)\\}$, then for any
$\phi\in\Lambda_{\mathcal{L}_{\alpha}}$ we have:

| $a\in l_{T}^{\alpha}(\phi)$
---|---
$\Leftrightarrow$ | $a^{*}\in l_{\mathfrak{A}}(\phi)$
$\Leftrightarrow$ | $h(a^{*})\in l_{\mathfrak{B}}(\phi)$
$\Leftrightarrow$ |  $\langle h(a^{*})(i_{1}),...,h(a^{*})(i_{k})\rangle\in\phi(\mathfrak{B})$ ($v_{i_{1}},...,v_{i_{k}}$ are free variables in $\phi$)
$\Leftrightarrow$ | $\langle b^{*}(\rho(i_{1})),...,b^{*}(\rho(i_{k}))\rangle\in\phi(\mathfrak{B})$
$\Leftrightarrow$ | $b^{*}\in l_{\mathfrak{B}}(\phi(v_{\rho(i_{1})}/v_{i_{1}})...(v_{\rho(i_{k})}/v_{i_{k}})))$
$\Leftrightarrow$ | $b\in l_{T}^{\alpha}(\phi(v_{\rho(i_{1})}/v_{i_{1}})...(v_{\rho(i_{k})}/v_{i_{k}})))$
$\Leftrightarrow$ | $b\in\rho l_{T}^{\alpha}(\phi)$

Hence $\rho:a\prec b$.

  * •

$3\Rightarrow 1:$ For any
$a\in|\mathcal{C}^{T}_{\alpha}|,b\in|\mathcal{C}^{T}_{\alpha}|$ representing
$\mathfrak{A},\mathfrak{B}$ respectively and any $\rho:a\prec b$, by Theorem
4.3, we know there are unique basis-preserving C-mappings
$g:(\mathcal{C}^{\mathfrak{A}}_{\alpha},l_{\mathfrak{A}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha}),g^{\prime}:(\mathcal{C}^{\mathfrak{B}}_{\alpha},l_{\mathfrak{B}})\rightarrow(\mathcal{C}^{T}_{\alpha},l_{T}^{\alpha})$
s.t. there are domain points $a^{*}\in g^{-1}[a],b^{*}\in g^{\prime-1}[b]$,
let $I:=\\{i\in\alpha:\mbox{ for any }j<i,a^{*}\in-D^{\mathfrak{A}}_{ij}\\}$,
clearly $A=\\{a^{*}(i):i\in I\\}$ and for any $i,j\in I$, $i\neq j\Rightarrow
a^{*}(i)\neq a^{*}(j)$. We define $B$ in the same way and define
$f:A\rightarrow B$ as: for any $i\in I$, $f(a^{*}(i))=b^{*}(\rho(i))$. Then
for any $\phi\in\Lambda_{\mathcal{L}_{\alpha}}$, $i_{1},...,i_{k}\in I$,

| $\langle a^{*}(i_{1}),...,a^{*}(i_{k})\rangle\in\phi(\mathfrak{A})$
---|---
$\Leftrightarrow$ | $a^{*}\in l_{\mathfrak{A}}(\phi)$
$\Leftrightarrow$ | $a\in l_{T}^{\alpha}(\phi)$
$\Leftrightarrow$ | $b\in\rho l_{T}^{\alpha}(\phi)$
$\Leftrightarrow$ | $b^{*}\in\rho l_{\mathfrak{B}}(\phi)$
$\Leftrightarrow$ | $b^{*}\in l_{\mathfrak{B}}(\phi(v_{\rho(i_{1})}/v_{i_{1}}))...(v_{\rho(i_{k})}/v_{i_{k}}))$