Abstract: We provide necessary and sufficient conditions for a correspondence
taking values in a finite-dimensional Euclidean space to be open so as to
revisit the pioneering work of Schmeidler (1969), Shafer (1974), Shafer-
Sonnenschein (1975) and Bergstrom-Rader-Parks (1976) to answer several
questions they and their followers left open. We introduce the notion of
separate convexity for a correspondence and use it to relate to classical
notions of continuity while giving salience to the notion of separateness as
in the interplay of separate continuity and separate convexity of binary
relations. As such, we provide a consolidation of the convexity-continuity
postulates from a broad inter-disciplinary perspective and comment on how the
qualified notions proposed here have implications of substantive interest for
choice theory. (119 words)

Journal of Economic Literature Classification Numbers: C00, C02, D01, D11,
D21, D51, D81

2020 Mathematics Subject Classification Numbers: 06A06, 26B05, 26B25, 91A44,
91B02, 91B08,

Key Words: Separately convex, separately continuous, linearly continuous,
graph-continuous, section-continuous, preferences, correspondences

Running Title: Continuity of Separately Convex Preferences

> Modo paulatim modo saltatim.111This is the epigraph of Gerschenkron (1962)
> and is difficult to translate by virtue of saltatim being associated with a
> dance metaphor, and paulatim a metaphor associated with gradualness. We are
> grateful to Richard Bett for his rendering, “little by little, jump by
> jump.”  Alexander Gerschenkron (1962)

> The definition of a “thing” is effected by means of continuity and of
> correlations which have a certain differential independence of other
> “things.”  Bertrand Russell (1915)222Address Delivered to the Philosophical
> Society of Manchester, in February, 1915 and reprinted as Chapter VII of
> Mysticism and Logic and Other Essays from the Monist, July, 1915. Russell
> explains: “That is to say, given a particular in one perspective, there will
> usually in a neighbouring perspective be a very similar particular,
> differing from the given particular, to the first order of small quantities,
> according to a law involving only the difference of position of the two
> perspectives in perspective space, and not any of the other ”things” in the
> universe. It is this continuity and differential independence in the law of
> change as we pass from one perspective to another that defines the class of
> particulars which is to be called “one thing.”

##  1 Introduction

In a remarkable paper, Gerschenkron (1962) writes, “The mathematician’s
continuity has the indubitable advantage of being an unambiguous concept, even
though it can be defined in a variety of more or less stringent fashions.” And
after furnishing a formal definition lifted from a then standard text in
mathematics, and another from one in mathematical economics,333See his
Footnote reference to Hyslop’s text, and also to R. G. D Allen’s tome on
Mathematical Economics. he continues as follows:

> This definition can be rendered even more simply by describing a continuous
> function as one that is “dense everywhere” in the sense of not having,
> strictly speaking, any contiguous points. For between any two points of such
> a function an infinite number of additional points can be placed. Stated
> still more simply, a continuous function is one that can be drawn in its
> entirety without lifting pencil from paper and which accordingly shows
> neither “gaps” nor “jumps.”

Everything is so neat and tidy that in his masterful essays, Koopmans (1957),
does not even see the need to define the concept,444See Footnote 4 in Uyanik
and Khan (2022) and the text it footnotes. The authors state “After a footnote
reference to a continuous function, an adjective he does not define but freely
uses, and then to a continuously representable preference ordering that he
does define, he masks the continuity postulate under the assumption of local
non-satiation, referring to the latter as “a rather weak continuity property
of preferences”. and as such, is at one with Gerschenkron. And so armed with
some clear validation and authority, the latter writes:

> Thus the historian who has gone out to scrutinize the mathematical concept
> of continuity travels far into strange lands and still is likely to return
> from his journey empty-handed. Continuity in this sense, or senses, does not
> appear to be a tool historians can profitably put to work; even when it
> appears in the empirical guise as gradualness of change it eludes the
> crucial problem of mensuration and is so thickly covered with the
> metaphysical paint of inevitability as to be destitute of all usefulness.

Fortunately, all this did not discourage the historian from launching his
classic in which five formulations of continuity are investigated for
philosophers and scientists, especially in the humanities.555This is not the
place to engage Gerschenkron (1962) but it is worthwhile to note his five
concepts of, historical change, they are “(a) constancy of direction; (b)
periodicity of events; (c) endogenous change; (d) length of causal regress;
(e) stability of the rate of change.”

This paper conducts an inquiry parallel to that of Gerschenkron, and its
results take as their point of departure the claim that the concept of
continuity has many meanings even when projected to the mathematical register.
In a series of recent papers, the authors go back to the foundational fact
established by Genocchi and Peano (1884) that “separate continuity does not
imply continuity,” and rely on it to provide a deconstruction and an
integration of the concept as it is used in economic theory.666The authors
note with some satisfaction the complementarity of Gerschenkron’s engagement
with Leibniz with the concluding discussion in Uyanik and Khan (2022). To be
sure, the fact that joint continuity is stronger than separate continuity was,
even then in the time of Cauchy in the early part of the 18th century, a
standard material in textbooks on multivariate calculus, and its investigation
constituted a rich development to which many mathematicians, including Heine,
Baire and Lebesgue. It culminated in the benchmark theorem of Rosenthal
(1955); see, for example, Ciesielski and Miller (2016) and the references
therein. How this work is then followed up and consolidated, specifically in
the context of both mathematical psychology and mathematical economics, see
Ghosh, Khan, and Uyanik (2022). The essential thrust of this work consisted of
two related analytical moves:

  * •

a translation of separate and linear continuity of functions to new continuity
concepts for correspondences and binary relations, and obtaining new results
on continuity of correspondences and binary relations, and

  * •

a consolidation, through this translation, of a variety of unrelated
continuity assumptions on binary relations dispersed in the economic
literature.

We give salience to convexity, and investigate in particular how notions of
“separate” convexity provide not only a consolidation but also answer several
questions left open in the pioneering papers. From an analytical point of
view, the marginal contribution of this paper then lies not so much in pushing
the internal investigation of the topological (continuity) register still
further, but in supplementing it with the sister-registers of linearity and
order. Thus it looks towards Hardy, Littelwood, and Pólya (1952(1934)), Young
(1910) and their followers for the impact on continuity of the convexity and
monotonicity postulates. Again, as in the previous inquiries, we take this
classical forking on functions and translate it to correspondence and
relations. Once redirected in this way, we are led naturally to separate as
opposed to joint (universal or global) convexity of sets and binary
relations.777The implications of separate convexity for distance functions in
optimization problems is studied in Bauschke and Borwein (2001).

Moving on to mainstream mathematical economics, we single out Schmeidler
(1969), Shafer (1974) and Bergstrom, Parks, and Rader (1976), among papers
that include Mas-Colell (1974), Gale and Mas-Colell (1975) and Shafer and
Sonnenschein (1975), to consider the relationship between section and graph
continuity assumptions for binary relations in mathematical economics. Under
completeness and transitivity, the two concepts are equivalent. Schmeidler
(1969) drops completeness but keeps transitivity and establishes the
equivalence under monotonicity. Shafer (1974) shows that under a strong
convexity assumption, the section and graph continuity postulates are
equivalent for a complete but non-transitive preference relation. Bergstrom,
Parks, and Rader (1976) provide generalizations of these results and a
comprehensive treatment on the relationship between section and graph
continuity postulates. In a recent paper, Gerasimou (2015) works with a
transitive and reflexive preference relation and establishes the equivalence
between section and graph continuity under the additivity assumption,888Even
though additivity is not a standard assumption in mathematical economics, it
and its related forms, have been used in decision theory; see Gerasimou (2010,
2013) for its use in the context of continuity of preferences. In a parallel
inquiry, Dubra, Maccheroni, and Ok (2004, Proposition 1) show that under the
independence assumption, (graph) mixture continuity is equivalent to graph
continuity for an incomplete preference relation, and Gilboa, Maccheroni,
Marinacci, and Schmeidler (2010) show that under the independence assumption,
reflexive and transitive weak preference relation has closed graph if and only
if it is mixture continuous. and the authors’ own work studies the
relationship among different continuity assumptions in mathematical economics
and decision theory under monotonicity or convexity of preferences.999See for
example, Uyanik and Khan (2022), Ghosh, Khan, and Uyanik (2022), Ghosh, Khan,
and Uyanik (2023), and their references. See also Karni (2007), Dubra (2011)
and Galaabaatar, Khan, and Uyanik (2019) on the relationship among scalar
continuity assumptions in decision theory.

This paper consists of three theorems that cover binary relations and
correspondences. Under separate convexity, Theorems 1 and 2 provide a
characterization of the open graph property for correspondences and restore
the equivalence between several notions of continuity of a correspondence.
Considering a binary relation as a graph of the correspondence, the two
theorems, along with their corollaries, substantially generalize the results
of Schmeidler (1969, 5.1), Shafer (1974, main Lemma) and Bergstrom, Parks, and
Rader (1976, Theorem 3) on the continuity of a binary relation by weakening
considerably their continuity and convexity/monotonicity assumptions, and by
allowing a more general domain on which the binary relation is defined.
Moreover, by weakening the convexity assumption, these results complement the
following quotation from Bergstrom, Parks, and Rader (1976):

> The assumption $P(x)\equiv\\{y\in\mathbb{R}^{n}:y\succ x\\}$ is convex
> demands more convexity than is needed for many purposes in general
> equilibrium analysis.

Furthermore, we answer in the negative the author’s question “whether
[Shafer’s result] can be generalized to infinite dimensional spaces or to an
arbitrary convex subset of $\mathbb{R}^{n}$.”101010We note in Section 4 how
our results can be generalized to an infinite dimensional setting. Note that
Yamazaki (1983) also partially answers this open problem question by providing
a generalization to infinite dimensional settings. He also claims in his
Corollary 3 that Shafer’s result holds also for arbitrary convex sets in
$\mathbb{R}^{n}$. Example 6 provides a counter-example to Yamazaki’s result.
In Theorem 3, under separate convexity, we provide a relationship among
various continuity assumptions in mathematical economics and mathematical
psychology for complete and transitive binary relations that is illustrated in
Figures 3 and 4.111111A detailed elaboration of this figure revolves around
how our results generalize antecedent results by (inessentially) weakening
continuity and (essentially) weakening the convexity assumption. For
elaboration on the notions of hiddenness and essentiality, deriving from Kim
and Richter (1986), see Uyanik and Khan (2022). In this connection, the reader
might also note the question of inconsistency of axioms as in Assa and Zimper
(2018). We also offer a host of examples to demonstrate the usefulness of our
results. Theorems 1–3 establish useful relationships between graph continuity
and other continuity postulates for non-ordered preferences, and by weakening
the convexity and monotonicity assumptions in earlier work, also generalize
it.

The paper is structured as follows. Section 2 presents the two main theorems
on the continuity of correspondences. Section 3 presents a portmanteau theorem
on the continuity of binary relations. Section 4 discusses some extensions of
these results and their applications to consumer and producer theory. Section
5 provides the required proofs.

##  2 On Continuity of Correspondences

In this section we present our two main theorems on the characterization of
open graph property that consolidate and generalize the existing results
presented in Schmeidler (1969), Shafer (1974) and Bergstrom, Parks, and Rader
(1976).

A correspondence from a topological space $X$ into a set
$Y\subseteq\mathbb{R}^{n}$ is a mapping $F:X\twoheadrightarrow Y$ that assigns
every $x\in X$ to a subset of $Y$. Define the graph of $F$ as gr$F=\\{(x,y)\in
X\times Y~{}|~{}y\in F(x)\\}$. For every $x\in X$, $F(x)$ denotes upper
section of $F$ at $x$, and for every $y\in Y$, $F^{-1}(y)=\\{x\in X~{}|~{}y\in
F(x)\\}$ denotes the lower section of $F$ at $y.$ $F$ has open sections if it
has both open upper and lower sections. A correspondence $F$ has separately
open upper sections if for every $x\in X$ and every straight line121212A
straight line in $Y\subseteq\mathbb{R}^{n}$ is the intersection of a one-
dimensional affine subspace of $\mathbb{R}^{n}$ and $Y$. A subset $X$ of a
(real) vector space is called affine if for all $x,y\in X$ and
$\lambda\in\mathbb{R},$ $\lambda x+(1-\lambda)y\in X.$ $L$ in $Y$ that is
parallel to a coordinate axis, $L\cap F(x)$ is open in $L$ and linearly open
upper sections if for every $x\in X$ and every straight line $L$ in $Y$,
$L\cap F(x)$ is open in $L$. It follows from their definitions that there is a
strict nested relationship between the following continuity postulates on a
correspondence $F$:

$\displaystyle\text{open graph}\Longrightarrow\text{open
sections}\Longrightarrow\text{linearly open
sections}\Longrightarrow\text{separately open sections.}$ (1)

A set $A\subseteq Y$ is separately convex if for every straight line $L$ in
$Y$ that is parallel to a coordinate axis $i=1,\ldots,n$, $A\cap L$ is convex.

Here we present our first result on graph continuity of correspondence defined
on a set that satisfies the following property.

(A) For a convex set $Y\subseteq\mathbb{R}^{n}$, either $Y$ is open or
$Y=\prod_{i=1}^{n}Y_{i}$, $Y_{i}\subseteq\mathbb{R}$ for all $i=1,\ldots,n$.

The following provides a partial converse relationship among the continuity
postulates listed in Equation 1.

######  Theorem 1.

Let $X$ be a topological space, $Y\subseteq\mathbb{R}^{n}$ satisfy property
LABEL:PrA, and $F:X\twoheadrightarrow Y$ has separately convex upper sections.
Then, $F$ has open graph if and only if it has open lower sections and
separately open upper sections.

Notice that the separate convexity assumption in Theorem 1 is weaker than
assuming that $F$ has convex values. Note that if we define a correspondence
$F:Y\twoheadrightarrow X$ from $Y$ into $X$ and define separately open lower
sections analogous to separately open upper sections, then we obtain the
following result that is symmetric to Theorem 1: if the lower sections of $F$
are separately convex, then $F$ has open graph if and only if it has open
upper sections and separately open lower sections. These two results imply the
following converse relationship among the continuity postulates listed in
Equation 1.

######  Corollary 1.

Let $X\subseteq\mathbb{R}^{n}$ satisfy property LABEL:PrA, and
$F:X\twoheadrightarrow X$ has separately convex upper sections and separately
convex lower sections. Then $F$ has an open graph if and only if it has
separately open lower sections and separately open upper sections.

Examples 1 and 2 are illustrations of the failure of the hypothesis in Theorem
1. While in Example 1, property LABEL:PrA fails, in Example 2, the separate
convexity assumption fails.

######  Example 1.

Let $X=\\{x\in[0,1]^{2}|~{}x_{2}\geqslant x_{1}\\}$,
$A=\\{x\in[0,1]^{2}|~{}x_{2}>x_{1}\\}\cup\\{(1,1)\\}\subset X$ and
$F:X\twoheadrightarrow X$ be a correspondence defined as $F(X)=A$ for all
$x\in X$. Note that $X$ is a bounded polytope, but is closed and hence fails
property LABEL:PrA. Further, $F$ has open lower sections and separately open
upper sections since for any straight line $L$ in $X$ that is parallel to a
coordinate axis, $L\cap A$ is open in $L$. However, the graph of $F$ is not
open since every neighborhood of $((1,1),(1,1))$ contains a point outside of
the graph of $F$.

######  Example 2.

Let $X=\mathbb{R}^{2}$,
$A=\mathbb{R}^{2}\backslash\\{x\in\mathbb{R}^{2}|x_{1}=x_{2},x\neq 0\\}$ and
$F:X\twoheadrightarrow X$ be a correspondence defined as $F(X)=A$ for all
$x\in X$. Clearly, $X$ satisfies property LABEL:PrA. Notice that $F$ is not
separately convex. The intersection of $A$ and any line parallel to a
coordinate axis is either the real line or a union of two open intervals.
Therefore, $F$ has separately open upper sections. However, it does not have
an open graph since every open ball containing the origin in $\mathbb{R}^{4}$
contains a point in the complement of the graph of $F$.

The next example gives two instances of non-convex preferences from Halevy,
Persitz, and Zrill (2017) that are separately convex.

######  Example 3.

Consider the following two preference relations that are defined on
$\mathbb{R}^{2}_{+}$ and are represented by the following utility functions:

$u_{1}(x,y)=\begin{cases}x^{3}y&\text{ if }x\geqslant y\\\ xy^{3}&\text{ if
}x\leqslant y\end{cases}$ (2)

and

$u_{2}(x,y)=\sqrt{\max\\{x,y\\}}+\dfrac{1}{4}\sqrt{\min{\\{x,y\\}}}.$ (3)

Figure 1: Utility functions in Equations (2) and (3).

In Figure 1, we illustrate the indifference curves corresponding to the two
utility functions above. Notice that while the preferences are not convex,
they are separately convex.

Our next result is on the continuity of correspondences defined on subspaces
that satisfy the following property.

(B) For a convex set $Y\subseteq\mathbb{R}^{n}$, either $Y$ is open or a
polyhedron, where a polyhedron is the intersection of a finite number of half-
spaces.

######  Theorem 2.

Let $X$ be a topological space, $Y$ a convex subset of $\mathbb{R}^{n}$
satisfies property LABEL:PrB and $F:X\twoheadrightarrow Y$ a correspondence
such that $F(x)$ is convex for all $x\in X$. Then, $F$ has an open graph if
and only if $F$ has open lower sections and linearly open upper sections.

Analogous to Theorem 1, if we define a correspondence $F:Y\twoheadrightarrow
X$ from $Y$ into $X$, then we obtain the following result that is symmetric to
Theorem 2: if the lower sections of $F$ are convex, then $F$ has open graph if
and only if it has open upper sections and linearly open lower sections. These
two results imply the following converse relationship among the first three
continuity postulates listed in Equation 1.

######  Corollary 2.

Let $X\subseteq\mathbb{R}^{n}$ satisfy property LABEL:PrB, and
$F:X\twoheadrightarrow X$ has convex upper sections and convex lower sections.
Then $F$ has open graph if and only if it has linearly open sections.

Note that unlike Theorem 1, open graph property in Theorem 2 is not equivalent
to open lower sections and separately open upper sections. For instance, $X$
is a bounded polytope in Example 1 and hence satisfies property B but the
correspondence $F$ fails to have linearly open upper sections, hence does not
have an open graph. Also notice that property LABEL:PrB is weaker than
property LABEL:PrA, however the convexity and continuity assumptions in
Theorem 2 are stronger than those in Theorem 1. Hence, the assumptions in
these two theorems are non-nested.

Example 2 above illustrates a correspondence that is separately convex and has
separately open upper sections but fails to have linearly open upper sections
as well as an open graph. The following example illustrates that if the
separate convexity assumption fails, then a correspondence may have both
separately open and linearly open upper sections but fails graph continuity.

######  Example 4.

Let $X=\mathbb{R}^{2}$,
$A=\mathbb{R}^{2}\backslash\\{x\in\mathbb{R}^{2}|x_{2}=x_{1}^{2},x\neq 0\\}$
and $F:X\twoheadrightarrow X$ be a correspondence defined as $F(X)=A$ for all
$x\in X$. Clearly, $X$ satisfies properties LABEL:PrA and LABEL:PrB but $F$ is
neither separately convex nor has convex upper sections. For any straight line
$L$, $L\cap A$ excludes at most two points of $L$. Hence, $F$ has both
separately open and linearly open upper sections. However, $A$ is not open
since every open ball containing $0$ contains a point in the complement of
$A$, hence $F$ fails to have open upper sections and an open graph.

The following example illustrates that the convexity assumption in Theorem 2
is essential even for a correspondence defined on an interval in $\mathbb{R}$.

######  Example 5.

Let $X=[0,1]$ and $F:X\twoheadrightarrow X$ such that $F(0)=(0,1]$ and
$F(x)=\\{y\in X~{}|~{}y>x,\text{ and }y\neq(1-x)\\}$ for all $x>0$. It is easy
to see that $F$ has open sections but $(0,1)$ has no open neighborhood
contained in the graph of $F$, hence $F$ does not have an open graph. It is
clear that $F(x)$ and $F^{-1}(y)$ are not convex for all $x\in(0,0.5)$ and for
all $y\in(0.5,1)$.

The next result provides additional relationships among the continuity
postulates.

######  Proposition 1.

Let $X$ be a topological space, $Y,Z$ two convex subsets of $\mathbb{R}^{n}$
that satisfy property LABEL:PrA, LABEL:PrB, respectively, and
$F:X\twoheadrightarrow Y,G:X\twoheadrightarrow Z$ be two correspondences.

  1. cmrmn(a)

if $F$ has separately convex upper sections, then the following are
equivalent: $F$ has (i) open upper sections, (ii) linearly open upper
sections, (iii) separately open upper sections.

  2. cmrmn(b)

if $G$ has convex upper sections, then $G$ has open upper sections if and only
if it has linearly open upper sections.

Note that the additional relationships among the lower continuity postulates
are obtained by suitably adjusting the domains and the ranges of the
correspondences and replacing “upper” in Proposition 1 by “lower.”

As in Theorem 1, considering a binary relation as the graph of a
correspondence, Theorem 2 provides necessary and sufficient conditions for a
binary relation on a convex subspace of $\mathbb{R}^{n}$ to be continuous. It
provides a characterization of the continuity of a binary relation, or a
correspondence, by using a topological property similar to the linear
continuity postulate. As noted in Equation 1, open upper (lower) sections
property is stronger than linearly open upper (lower) sections property that
is stronger than separately open upper (lower) sections property. Therefore,
Theorems 1 and 2 generalize the following result of Shafer (1974, Lemma, pg.
914) and Bergstrom, Parks, and Rader (1976, Theorem 3) on continuity of a
binary relation by considerably weakening their continuity and convexity
assumptions, and allowing a more general domain on which the binary relation
is defined. Define a binary relation $P$ on $X,$ where $P\subseteq X\times X.$
Note that for any binary relation $P$ on $X$, there exists a unique
correspondence $F:X\twoheadrightarrow X$ such that $P=\text{gr}F$. The upper
and lower sections of a binary relation $P$ at $x\in X$ is defined as
$P(x)=F(x)$ and $P^{-1}(x)=F^{-1}(x)$, respectively.

######  Corollary 3.

Let $X=\mathbb{R}^{n}_{+}$ and $P:X\twoheadrightarrow X$ be a correspondence
such that $P(x)$ is convex for all $x\in X$ (or $P^{-1}(x)$ is convex for all
$x\in X$). Then, $P$ has an open graph if and only if $P$ has open sections.

Moreover, along with Theorems 1 and 2, Corollary 1 generalize the following
result of Schmeidler (1969) by dropping the transitivity assumption,
substantially weakening the continuity and monotonicity assumptions,131313It
is easy to observe that if a strict binary relation is strongly monotone, then
it has separately convex upper and lower sections, but the converse
relationship does not hold as for example the relation need not be complete. A
similar relationship holds for weak monotonicity whose proof is provided in
Section 5. and allowing a more general domain on which the binary relation is
defined.

######  Corollary 4.

Let $X=\mathbb{R}^{n}_{+}$ and $P:X\twoheadrightarrow X$ be a transitive,
irreflexive and strongly monotone binary relation on $X$. Then $P$ has an open
graph if and only if $P$ has open sections.

Note that Gerasimou (2015) provides a result on the relationship between
sections and graph continuity of a reflexive and transitive binary relation
under the additivity assumption. The results in this paper focus on separate
convexity and do not impose additivity. It is easy to show that additivity and
separate convexity assumptions are non-nested for preferences defined in this
paper and in Gerasimou (2015). Therefore, our results and the results
presented in Gerasimou (2015) are non-nested.

Bergstrom, Parks, and Rader (1976, Theorem 1) provides a generalization of
Schmeidler’s result for a transitive and order-dense binary relation on
general topological spaces. Their result and our results above are non-nested;
while we impose weaker continuity assumptions and do not assume transitivity
or order-denseness, they do not impose any convexity assumption and allow for
a more general domain. Further, they note that the generalization of Theorem 2
to arbitrary convex sets in $\mathbb{R}^{n}$ is an open problem. In Example 6,
we answer this open problem in the negative and also provide a counterexample
to Corollary 3 of Yamazaki (1983) on the open problem.141414Yamazaki (1983,
Proposition 2) identifies the following property for a subspace of a
topological vector space: a subspace $X$ of a topological vector space is
locally finite if for each $z\in X$ there exists a finite collection of points
$\\{x^{1},\ldots,x^{k}\\}$ in $X$ such that the convex hull of
$\\{x^{1},\ldots,x^{k}\\}$ is a neighborhood of $z$. Yamazaki’s Proposition 2
provides a generalization of Shafer’s result to spaces satisfying this
property. (He also works with a weakening of convex domains by restricting the
preferences to the restricted domain). The local finiteness property has its
roots in the proof of Shafer’s (1974) result and plays a crucial role.
However, Yamazaki’s application of his Proposition 2 to solve the open problem
mentioned above has a mistake.  The example further demonstrates that the
results in this section cannot be generalized to a setting where $Y$ is an
arbitrary convex set in $\mathbb{R}^{n}$.

Figure 2: A convex binary relation with open sections whose graph is not open

######  Example 6.

Let $X=\\{x\in\mathbb{R}_{+}^{2}|x_{1}^{2}+x_{2}^{2}\leqslant 1\\}$. Clearly,
$X$ is convex. Consider a homeomorphism
$f:[0,1]\rightarrow\\{x\in\mathbb{R}_{+}^{2}|x_{1}^{2}+x_{2}^{2}=1\\}$ as
illustrated in Figure 2 where $f(0)=(0,1)$ and $f(1)=(1,0)$. Define a
correspondence $F:X\twoheadrightarrow X$ as follows: $F(1,x_{2})=X$ for all
$(1,x_{2})\in X$ and $F(x_{1},x_{2})=X\backslash\\{f(x_{1})\\}$ for all $x\in
X$ with $x_{1}\neq 1$. Note that $F$ is constant in the second variable.
Moreover, $F(x)$ is convex for all $x\in X$ since it excludes at most one
point that cannot be written as a weighted average of two points in $F(x)$.
Also, $F(x)$ is open in $X$ for all $x\in X$ since its complement is closed
(either an empty set or a singleton). Furthermore,

$F^{-1}(y)=\begin{cases}X&\text{ if $y\notin f([0,1))$ }\\\
X\backslash(\\{a\\}\times\\{z_{2}\in\mathbb{R}_{+}~{}|~{}(a,z_{2})\in
X\\})&\text{ if }y=f(a)\text{ for some }a\in[0,1).\end{cases}$

Note that for each $y\in\\{x\in\mathbb{R}_{+}^{2}|x_{1}^{2}+x_{2}^{2}=1\\}$,
$y\neq(1,0)$, there exists a unique such $a\in[0,1)$ such that $y=f(a)$.
Hence, $F$ has open lower sections. However, $F$ does not have an open graph
since every neighborhood of $((1,0),(0,1))\in\text{gr}F$ contains a point in
the complement of gr$F$. In this example, setting $P=\text{gr}F$ implies that
$P\subseteq X\times X$ is a binary relation on the convex set $X$ with open
sections and convex values. However, $P$ is not open in $X\times X$.
$\square$

The results we present here either focus on correspondences with open graph,
or on open binary relations.151515In a related setting, Zhou (1995,
Proposition 2) replaces open lower sections assumption in Shafer’s result with
lower semicontinuity of a correspondence whose values are in
$Y=\mathbb{R}^{n}$ to provide a characterization of the open graph property;
see also Impicciatore and Ruscitti (2012) for a recent treatment. By using the
arguments in this paper, it is possible to show that the open and convex upper
sections in Zhou’s result can be replaced with separately open and separately
continuous upper sections.  Example 7 illustrates that a binary relation need
not be closed even if it has closed sections and convex upper sections. There
are results in the literature that show equivalence between having closed
sections and being closed for a binary relation; see for example, Ward (1954)
and Shafer (1974) under completeness or transitivity assumptions.

######  Example 7.

Let $X=[0,1]$ and $P\subseteq X\times X$ such that $P(x)=\\{x\\}$ for all
$x<1$ and $P(1)=\\{0\\}$. It is clear that $P$ has closed sections (both upper
and lower). Also, $P(x)$ is convex-valued since it’s a singleton for all $x\in
X$. However, $P$ is not closed in $X\times X$ since $(1,1)\in P^{c}$ has no
open neighborhood contained in $P^{c}$.

##  3 On Continuity of Ordered Preferences

In the previous section, we provide relationships among different continuity
assumptions under separate convexity assumption for a binary relation that
need not be complete or transitive. In this section, we bring out the
additional relationships when the binary relation is ordered, that is,
complete and transitive.

Let $X\subseteq\mathbb{R}^{n}$ be a convex set. A subset $\succsim$ of
$X\times X$ denotes a binary relation on $X.$ We denote an element
$(x,y)\in~{}\\!\\!\\!\succsim$ as $x\succsim y.$ The asymmetric part $\succ$
of $\succsim$ is defined as $x\succ y$ if $x\succsim y$ and $y\not\succsim x$,
and its symmetric part $\sim$ is defined as $x\sim y$ if $x\succsim y$ and
$y\succsim x.$ The inverse of $\succsim$ is defined as $x\precsim y$ if
$y\succsim x$. Its asymmetric part $\prec$ is defined analogously and its
symmetric part is $\sim$. For any $x\in X$, let $A_{\succsim}(x)=\\{y\in
X|y\succsim x\\}$ denote the upper section of $\succsim$ at $x$, and
$A_{\precsim}(x)=\\{y\in X|y\precsim x\\}$ its lower section at $x$. For any
$x,y\in X$ and $\lambda\in[0,1]$, let $x\lambda y$ denote $\lambda
x+(1-\lambda)y$. We provide in Table 1 the descriptive adjectives pertaining
to a relation in a tabular form for the reader’s convenience.161616For vectors
$x$ and $y$, “$x\geqslant y$” means $x_{i}\geqslant y_{i}$ in every component;
“$x>y$” means $x\geqslant y$ and $x\neq y$; and “$x\gg y$” means $x_{i}>y_{i}$
in every component.

reflexive | $x\succcurlyeq x,\ \forall x\in X$
---|---
non-trivial |  $\exists x,y\in X$ such that $x\succ y$
complete |  $x\succcurlyeq y$ or $y\succcurlyeq x,\ \forall x,y\in X$
transitive | $x\succcurlyeq y\succcurlyeq z\Rightarrow x\succcurlyeq z,\ \forall x,y,z\in X$
monotone |  $x>y\implies x\succsim y\ \forall\ x,y\in X$ or  $x>y\implies x\precsim y\ \forall\ x,y\in X$
order dense | $x\succ y\implies\exists z\in X\ \mbox{such that}\ x\succ z\succ y\ \forall\ x,y\in X$

Table 1: Properties of binary relations

The continuity assumption on a binary relation is one of the standard
assumptions in decision theory and mathematical psychology. The following
definition lists the many forms it takes in the literature.

######  Definition 1.

A binary relation $\succsim$ defined on a convex set
$X\subseteq\mathbb{R}^{n}$ is

  1. cmrmn(a)

graph continuous if $\succsim$ is a closed subset of $X\times X$.

  2. cmrmn(b)

upper continuous if $\succsim$ has closed upper sections, lower continuous if
$\succsim$ has closed lower sections, and continuous if $\succsim$ has closed
upper and lower sections.

  3. cmrmn(c)

linearly upper (lower) continuous if the restriction of the upper (lower)
sections of $\succsim$ to any straight line $L$ in $X$ is closed in $L$, and
linearly continuous if $\succsim$ is linearly upper and linearly lower
continuous.

  4. cmrmn(d)

separately upper (lower) continuous if the restriction of the upper (lower)
sections of $\succsim$ to any straight line $L$ in $X$ that is parallel to a
coordinate axis is closed in $L$, and separately continuous if $\succsim$ is
separately upper and separately lower continuous.

  5. cmrmn(e)

upper (lower) mixture continuous if for any $x,y,z\in X$,
$\\{\lambda\in[0,1]|x\lambda y\succsim z\\}$ is closed
($\\{\lambda\in[0,1]|x\lambda y\precsim z\\}$ is closed) in the unit interval
$[0,1]$, and mixture continuous if $\succsim$ is upper mixture and lower
mixture continuous.

  6. cmrmn(f)

upper $\left(\text{lower}\right)$ Archimedean if for any $x,y,z\in X,x\succ y$
implies that there exists $\lambda\in(0,1)$ $\left(\delta\in(0,1)\right)$ such
that $x\lambda z\succ y$ $\left(x\succ y\delta z\right)$, and Archimedean if
it is upper and lower Archimedean.

  7. cmrmn(g)

Wold-continuous if it is order-dense and $x\succ z\succ y$ implies that any
curve171717A curve on $X$ is the image of a continuous injective function
$m:[0,1]\rightarrow X$. joining $x$ to $y$ meets the indifference class of
$z$.

  8. cmrmn(h)

weakly Wold-continuous if it is order-dense and $x\succ z\succ y$ implies that
the straight line joining $x$ to $y$ meets the indifference class of $z$.

  9. cmrmn(i)

restricted solvable if for all $i\in\\{1,\ldots,n\\},$ all $x,y\in X$ and all
$(a_{i},y_{-i}),(b_{i},y_{-i})\in X$ with $(a_{i},y_{-i})\succsim
x\succsim(b_{i},y_{-i})$, there exists $c\in X$ with $(c_{i},y_{-i})\in X$
such that $x\sim(c_{i},y_{-i}).$

Moreover, $\succsim$ possesses the

  1. cmrmn(j)

intermediate value property (IVP) if for all $x,y,z\in X$ with $x\succsim
y\succsim z$, there exists $\lambda\in[0,1]$ such that $x\lambda z\sim y$.

  2. (k)

strong IVP if for all $x,y,z\in X$ with $x\succsim y\succsim z$ and all curves
$C_{xy}$ connecting $x$ and $z$, there exists $c\in C_{xy}$ such that $c\sim
y$.

In his pioneering work, Wold (1943–44) developed consumer theory and provided
a result on numerical representation of binary relations by a continuous
function,181818Wold’s representation theorem and the well-known representation
theorem of Eilenberg (1941) are independent discoveries, and the proofs are
different; see Banerjee and Mitra (2018), Uyanik and Khan (2022) and Ghosh,
Khan, and Uyanik (2023) for details. based on the Wold continuity assumption.
In subsequent work, Wold and Jureen (1953) uses a weaker version of Wold
continuity, the weak-Wold assumption. This weaker continuity assumption is
first used in decision theory independently by Nash (1950) and Marschak (1950)
who provide a complete axiomatization of the expected utility theory initiated
by von Neumann and Morgenstern (1947).191919von Neumann and Morgenstern (1947)
use a version of the Archimedean assumption in their theory. There are many
versions of the axiomatics of the expected utility theory. Nash and Marschak
use weak Wold-continuity while Herstein and Milnor (1953) use mixture
continuity; see Bleichrodt, Li, Moscati, and Wakker (2016) for a history of
axiomatics of the expected utility theory. These Wold-continuity postulates
are analogous to restricted solvability and the two IVP postulates. The linear
and separate continuity postulates are motivated by their classical
counterparts for functions in mathematics. Note that linear continuity is
equivalent to mixture continuity. The remaining continuity postulates are
standard in mathematical economics and decision theory.

######  Theorem 3.

Let $\succsim$ be a complete and transitive binary relation on a non-empty and
convex set $X\subseteq\mathbb{R}^{n}$ with property LABEL:PrA.

  1. cmrmn(a)

If the upper (lower) sections of $\succsim$ are separately convex, then the
following are equivalent for $\succsim$: upper (lower) continuity, upper
(lower) mixture continuity, upper (lower) linear continuity, upper (lower)
separate continuity, lower (upper) Archimedean.

  2. cmrmn(b)

If $\succsim$ is order dense and its sections are separately convex, then the
following are equivalent for $\succsim$: graph continuity, continuity, linear
continuity, mixture continuity, Archimedean, strong IVP, IVP, Wold-continuity,
weak Wold-continuity, restricted solvability, and separate continuity.

The results in the previous section provide equivalence relationship among
separate continuity, linear continuity, continuity, and graph continuity of a
preference relation that need not be complete or transitive. Theorem 3(a) adds
Archimedean and mixture continuity into this relationship when the upper or
the lower sections of the relation are separately convex. Part (b) of the
theorem shows that if the separate convexity assumption holds for both upper
and lower sections, then further equivalence relationships hold among the
defined continuity postulates.

Note that the convexity assumption in part (b) of the theorem is equivalent to
the following monotonicity property: $\succsim$ is separately monotone if
$x_{i}>y_{i}$ implies $(x_{i},z_{-i})\succsim(y_{i},z_{-i})$ for all
$i=1,\ldots,n$ and all $(x_{i},z_{-i}),(y_{i},z_{-i})\in X$, or $x_{i}>y_{i}$
implies $(x_{i},z_{-i})\succsim(y_{i},z_{-i})$ for all $i=1,\ldots,n$ and all
$(x_{i},z_{-i}),(y_{i},z_{-i})\in X$. Moreover, it is not difficult to show
that a transitive binary relation $\succsim$ is separately monotone if and
only if it is monotone. We provide proof of these claims in the next section.
Ghosh, Khan, and Uyanik (2022) provide an equivalence relation among different
continuity postulates under monotonicity and convexity. Theorem 3(b) provides
a generalization by allowing a more general domain and an alternative proof of
their result and connects it to separate convexity.

Figures 3 and 4 provide pictorial representations of the results we report in
Theorem 3. The reader should note in particular how the notion of separate
convexity allows the execution and the completion of several equivalences.

Figure 3: Relationship among continuity postulates in Theorem 3(a). Figure 4:
Relationship among continuity postulates in Theorem 3(b).

######  Example 8.

This example illustrates that separate convexity is not enough to obtain
equivalence between Archimedeanity and mixture continuity and between separate
continuity and continuity as well as restricted solvability and continuity. It
is motivated by the classic counterexample of Genocchi and Peano
(1884).202020There is a rich and deep literature on the relationship among
different continuity assumptions on functions; see for example, Ciesielski and
Miller (2016) and Uyanik and Khan (2022) for a detailed discussion. Let
$X=\mathbb{R}^{2}$ and $f:X\rightarrow\mathbb{R}$ defined as follows:

$f(x_{1},x_{2})=\frac{2x_{1}x_{2}}{x_{1}^{2}+x_{2}^{2}}+\mbox{
min}\\{x_{1},x_{2}\\}\text{ if
}x\in\mathbb{R}_{+}^{2}\backslash\\{0,0\\},\text{ and }f(x)=0\text{
otherwise}.$

Let $\succsim$ be the binary relation on $X$ induced by $f$, that is
$x\succsim y$ if and only if $f(x)\geqslant f(y)$. In this case, it is easy to
show that $\succsim$ is separately convex and satisfies Archimedean property,
separate continuity and restricted solvability. However, $\succsim$ fails
mixture continuity along the $45^{\circ}$ line, hence also fails continuity.

##  4 Discussion and Remarks

In this section, we list three observations about our results. First, the
results here (Theorems 1 and 2) can be generalized to a topological vector
space setting by using an argument analogous to those in the proofs. Let
$X_{0}$ be an arbitrary non-empty topological space and
$\\{X_{i}\\}_{i=1}^{n}$ be a finite collection of non-empty, convex finite
sets in a topological vector space.212121See Footnote 14 for the definition of
a locally finite set. Let $A\subseteq\prod_{i=0}^{n}X_{i}$ be a set such that
for all $i=1,\ldots,n$ and all $x\in A$, $A_{i}(x)$ is convex. Then, $A$ is
open if and only if $A_{i}(x)$ is open for all $i=0,1,\ldots,n$ and all $x\in
A$. Note that $A$ can be considered as the graph of correspondence from
$X_{0}$ into $X=\prod_{i=1}^{n}X_{i}$, or as a binary relation on $X$ when
$X_{0}=X$. Moreover, generalizations of the results (Theorem 3) concerning the
relationship between different continuity postulates for functions can also be
obtained.

Second, a major thrust of our paper has been to explore the notion of
separateness within the choice theory register. Extending this analysis to
producer theory appears to be a promising avenue for future work. In this
context, Herberg (1973) develops a notion of point-wise convexity, similar to
the idea of separate convexity in this paper.222222See for example Figures 1–4
and Lemma 1 of Herberg (1973). As another simple example, the preference
relation induced by the utility function $\max\\{x_{1},x_{2}\\}$ is separately
convex but not convex, in fact, its lower sections are convex. Note that the
production set induced by the function above is convex. To the best of our
knowledge, there is no work that has explored this issue further. Moreover, a
weaker convexity assumption on production sets may give us a sufficient
background to work with marginal rates.

Finally, non-convexity in preferences continues to be an important avenue for
exploration.232323Trockel (1984) writes on the role of non-convexity in the
context of market demand and large economies. Recent work by Halevy, Persitz,
and Zrill (2017) suggests useful implications of non-convexity for techniques
in revealed preferences. We hope that by bringing separate convexity to the
picture, we can stimulate applications in this area.

##  5 Proofs of the Results

###### Proof of Theorem 1.

Let $X$ be a topological space, $Y\subseteq\mathbb{R}^{n}$ be a convex set
with property LABEL:PrA, $F:X\twoheadrightarrow Y$ be a correspondence with
separately convex upper sections and $I=\\{1,\ldots,n\\}$. For notational
simplicity, define $A\subseteq X\times Y$ as $A=\text{gr}F$. For all $(x,y)\in
A$, let $A_{0}(x,y)=\\{x^{\prime}\in X|~{}(x^{\prime},y)\in A\\}$, and for any
$J\subseteq I$, let $A_{J}(x,y)=\\{y^{\prime}\in
Y|~{}y^{\prime}_{-J}=y_{-J}\text{ and }(x,y^{\prime})\in A\\}$ and
$A(y_{J})=\\{(x^{\prime},y^{\prime})\in A~{}|~{}y^{\prime}_{J}=y_{J}\\}$. The
forward direction is clear. For the backward direction, let $Z=X\times Y$, and
assume $F$ has open lower sections and separately open upper sections, that is
$A_{0}(z)$ is open and $A_{I}(z)$ is separately open for all $z\in A$.

Next, we show that for all $J\subsetneq I$ and all $(x,y)\in A$, if
$A(y_{-J})$ is open in $Z(y_{-J})$, then $A(y_{-(J\cup\\{i\\})})$ is open in
$Z(y_{-(J\cup\\{i\\})})$ for each $i\notin J$. Towards this end, pick
$J\subsetneq I$, $(\bar{x},\bar{y})\in A$ and assume $A(\bar{y}_{-J})$ is open
in $Z(\bar{y}_{-J})$. Pick
$z=(x,y_{J\cup\\{i\\}},\bar{y}_{-(J\cup\\{i\\})})\in A$ and a straight line
$L_{y_{i}}$ in $Y$ that contains
$(y_{J\cup\\{i\\}},\bar{y}_{-(J\cup\\{i\\})})$ and is parallel to the $i$-th
coordinate axis in $Y$. Since $A_{I}(z)$ is separately open in $Y$, there
exist $a_{i},b_{i}\in\mathbb{R}$, $a_{i}\leqslant b_{i}$, such that
$U=[a_{i},b_{i}]\times\\{y_{J},\bar{y}_{-(J\cup\\{i\\})}\\}$ is contained in
$L_{y_{i}}\cap A_{I}(z)$ and is a neighborhood242424A set is a neighborhood of
a point in a topological space if it contains the point in its topological
interior. of $(y_{i},y_{J},\bar{y}_{-(J\cup\\{i\\})})$ in the subspace
$L_{y_{i}}$.

Since $(x,\tilde{y}_{i},y_{J},\bar{y}_{-(J\cup\\{i\\})})\in A$ for all
$\tilde{y}_{i}\in[a_{i},b_{i}]$, the sets $A(a_{i},\bar{y}_{-(J\cup\\{i\\})})$
and $A(b_{i},\bar{y}_{-(J\cup\\{i\\})})$ are open in
$Z(a_{i},\bar{y}_{-(J\cup\\{i\\})})$ and $Z(b_{i},\bar{y}_{-(J\cup\\{i\\})})$
respectively, and contain $(x,y_{J})$, there exist neighborhoods
$U^{a}(x,y_{J})\subseteq A(a_{i},\bar{y}_{-(J\cup\\{i\\})})$ and
$U^{b}(x,y_{J})\subseteq A(b_{i},\bar{y}_{-(J\cup\\{i\\})})$ of $(x,y_{J})$.
Let $U(x,y_{J})=U^{a}(x,y_{J})\cap U^{b}(x,y_{J})$. Then, $U(x,y_{J})$ is a
neighborhood of $(x,y_{J})$ and for all $(x^{\prime},y^{\prime}_{J})\in
U(x,y_{J}),(x^{\prime},y^{\prime}_{J})\in U^{a}(x,y_{J})\subseteq
A(a_{i},\bar{y}_{-(J\cup\\{i\\})})$ and $(x^{\prime},y^{\prime}_{J})\in
U^{b}(x,y_{J})\subseteq A(b_{i},\bar{y}_{-(J\cup\\{i\\})})$. Hence for all
$(x^{\prime},y^{\prime}_{J})\in U(x,y_{J})$, $a_{i},b_{i}\in
A(x^{\prime},y_{J}^{\prime},\bar{y}_{-(J\cup\\{i\\})})$. Then, it follows from
the convexity assumption that for all $(x^{\prime},y^{\prime}_{J})\in
U(x,y_{J})$, $[a_{i},b_{i}]\subseteq
A(x^{\prime},y_{J}^{\prime},\bar{y}_{-(J\cup\\{i\\})})$. Note that
$[a_{i},b_{i}]\times U(x,y_{J})\times\\{\bar{y}_{-(J\cup\\{i\\})}\\}\subseteq
A$ and it is a neighborhood of $z$ in the subspace
$Z(\bar{y}_{-(J\cup\\{i\\})})$. Therefore, $A(\bar{y}_{-(J\cup\\{i\\})})$ is
open in $Z(\bar{y}_{-(J\cup\\{i\\})})$

Note that $A_{0}(z)$ is open in $X$ for all $z\in A$. By setting
$J=\emptyset$, it follows from the argument above that for all $(x,y)\in X$,
$A(y_{-1})$ is open in $Z(y_{-1})$. Iteratively adding one index
$i=2,\ldots,n$ into $J$ and applying the argument above imply that $A$ is open
in $Z$.

Let $X$ be a subset of $\mathbb{R}^{n}.$ The closure of $X$ is denoted by
cl$X$ and its interior by int$X$. Since any lower dimensional subset of
$\mathbb{R}^{n}$ has an empty interior, it is more convenient to work with the
concept of relative interior. Recall that a subset $X$ of a (real) vector
space is affine if for all $x,y\in X$ and $\lambda\in\mathbb{R},$ $\lambda
x+(1-\lambda)y\in X.$ It is clear that $A$ is affine if and only if
$A-\\{a\\}$ is a subspace of $X$ for all $a\in A.$ The affine hull of $X,$
aff$X,$ is the smallest affine set containing $X$. The relative interior of a
subset $X$ of $\mathbb{R}^{n}$ is defined as

$\text{ri}X=\\{x\in\mbox{aff}X~{}|~{}\exists N_{\varepsilon},\mbox{ an
}\varepsilon\text{ neighborhood of }x,\text{ such that
}N_{\varepsilon}\cap\text{aff}X\subseteq X\\}.$

That is, the relative interior of $X$ is the interior of $X$ with respect to
the smallest affine subspace containing $X$. The following result is due to
Rockafellar (1970, p. 45).

######  Lemma 1.

Let $X$ be a non-empty and convex subset of $\mathbb{R}^{n}.$ Then
$\mbox{ri}X$ is non-empty, and for all $x\in\mbox{ri}X,y\in\mbox{cl}X$ and all
$\lambda\in[0,1),$ $y\lambda x\in\mbox{ri}X.$

###### Proof of Theorem 2.

The forward direction is obvious. For the backward direction, assume
$F:X\twoheadrightarrow Y$ has convex and linearly open upper sections and open
lower sections. First, we show that $F$ has open upper sections, that is
$F(x)$ is open for all $x\in X$. Assume towards a contradiction that there
exists $x_{0}\in X$ such that $F(x_{0})$ is not open (in $Y$), hence there
exists $y\in F(x_{0})$ such that $y$ is not an interior point of $F(x_{0})$.
Note that $F(x_{0})$ is convex and has infinitely many elements. Let
$\mathcal{H}_{y}$ denote the set of all supporting hyperplanes of $F(x_{0})$
at ${y}$ in the affine space generated by $F(x_{0})$, and for all
$h\in\mathcal{H}_{y}$, let $H_{h}$ denote the closed half-space determined by
$h$ that contains $F(x_{0})$.

Assume there exists $h\in\mathcal{H}_{y}$ such that $H^{c}_{h}\cap
F(x_{0})\neq\emptyset$. Since $H^{c}_{h}\cap F(x_{0})$ is non-empty and
convex, pick $w$ in the relative interior of $H^{c}_{h}\cap F(x_{0})$, which
is nonempty by Lemma 1. Let $L_{yw}$ be the straight line in $Y$ that contains
$y$ and $w$. It follows from Lemma 1 that $y\delta w\notin F(x_{0})$ for all
$\delta\in[0,1)$. Then, every open neighborhood of $y$ in the subspace
$L_{yw}$ contains an element $y\delta w\notin F(x_{0})$. Since $y\in F(x_{0})$
this yields a contradiction with linear openness of $F(x_{0})$.

Next, assume for all $h\in\mathcal{H}_{y}$, the set $H^{c}_{h}\cap
F(x_{0})=\emptyset$. This case happens only if $Y$ is a polyhedron and $y$
lies in one of the hyperplanes determining the relative boundary of $Y$
defined in property LABEL:PrB. Let $\mathcal{H}^{\prime}_{y}$ be the set of
all such hyperplanes. If there exists $h\in\mathcal{H}^{\prime}_{y}$ and
$y^{\prime}\in h$ such that for all $\delta\in[0,1)$, $y\delta
y^{\prime}\notin F(x_{0})$, then this yields a contradiction with the linear
openness of $F(x_{0})$. Otherwise, if for all $h\in\mathcal{H}^{\prime}_{y}$
and all $y^{\prime}\in h$, there exists $\delta\in[0,1)$ such that $y\delta
y^{\prime}\in F(x_{0})$, then there exist finitely many points in these
finitely many hyperplanes such that the convex hull of these points constitute
a (closed) neighborhood of $y$. Hence $y$ is an interior point of $F(x_{0})$
which yields a contradiction with the assumption that $y$ is not an interior
point of $F(x_{0})$. Therefore, $F(x)$ is open for all $x\in X$.

It remains to show that gr$F$ is open. Towards this end, pick
$z_{0}=(x_{0},y_{0})\in\text{gr}F$. Since $F$ has open upper sections and $Y$
satisfies property LABEL:PrB, there exist finitely many points
$p^{1},\ldots,p^{m}$ in $F(x_{0})$ such that
$V=\text{co}\\{p^{1},\ldots,p^{m}\\}$ is a neighborhood of $y_{0}$ in $Y$.

For each $k=1,\ldots,m$, we have $p^{k}\in F(x_{0})$, hence $x_{0}\in
F^{-1}(p^{k})$. For each $k=1,\ldots,m$, since $F^{-1}(p^{k})$ is open, there
exists a neighborhood $U^{k}$ of $x_{0}$ in $X$ such that $U^{k}\subseteq
F^{-1}(p^{k})$. Then, the set $U=\bigcap_{k=1}^{m}U^{k}$ is a neighborhood of
$x_{0}$ and $U\subseteq U^{k}\subseteq F^{-1}(p^{k})$ for all $k$. Hence, for
all $x\in U$ and all $k=1,\ldots,m$, $p^{k}\in F(x)$. For all $x\in U$, since
$F(x)$ is convex, $V=\text{co}\\{p^{1},\ldots,p^{m}\\}\subseteq F(x)$. Then,
for all $x\in U$ and all $y\in V$, $x\in F^{-1}(y)$ and $y\in F(x)$.
Therefore, $U\times V$ is a neighborhood of $z_{0}$, hence $z_{0}$ is an
interior point of gr$F$.

Since a subset of the product space $X\times Y$ can be defined as the graph of
a correspondence from $X$ into $Y$, Theorems 1 and 2 provide characterizations
of open sets when the sections of the set satisfy a suitable convexity
assumption; see for example Halkin (1966), Uyanik and Khan (2023) and their
references for characterizations of open sets in $\mathbb{R}^{n}$, and also
see Fan (1966) for applications of sets with convex sections.

###### Proof of Proposition 1.

Part (b) directly follows from the argument in the proof of Theorem 2 above as
the first step of the proof shows that under convex upper sections, linearly
open upper sections is equivalent to open upper sections. To prove part (a),
assume $F$ has separately convex and separately open upper sections. Pick
$x\in X$. We need to show that $F(x)$ is open. If $Y$ is a product set, then
define a correspondence $H:Y_{i}\twoheadrightarrow Y_{-i}$ for some
$i=1,\ldots,n$ such that gr$H=F(x)$. Then, by Theorem 1, $H$ has an open graph
in the subspace $Y$, hence $F(x)$ is open in $Y$. If $Y$ is an open set in
$\mathbb{R}^{n}$, then define a correspondence
$G:\mathbb{R}\twoheadrightarrow\mathbb{R}^{n-1}$ such that gr$G=F(x)$.
Analogously, by Theorem 1, $G$ has an open graph in $\mathbb{R}^{n}$. Since
$Y$ is open, therefore $F(x)$ is open in $Y$.252525Note that the convexity of
the upper sections restricted to lines parallel to coordinate axis $i$ is not
needed in part (a). For clarity of the exposition, we do not relax this
assumption in the statement of the proposition.

###### Proof of Theorem 3.

The one-directional strict inclusion relationship among the continuity
postulates does not require a convexity assumption and is provided in the
antecedent literature; see Ghosh, Khan, and Uyanik (2022, Proposition 9) for
details and references.

In part (a), the statement that under the suitable separate convexity
assumption, separate upper (lower) continuity implies upper (lower) continuity
follows from Proposition 1(a). Therefore, showing upper (lower) Archimedean
implies lower (upper) separate continuity under the suitable separate
convexity assumption completes the proof of part (a). Towards this end, assume
$\succsim$ has separately convex upper sections, satisfies lower Archimedean
property but fails to have separately closed upper sections. Hence, there
exists $x\in X$ and a line $L$ parallel to a coordinate axis such that
$A_{\succsim}(x)\cap L$ is not closed. Then, there exist $x\in X$, an index
$i$, and a sequence $(y^{k}_{i},\bar{y}_{-i})\rightarrow(y_{i},\bar{y}_{-i})$
on the line $L_{i}$ parallel to coordinate $i$ such that
$(y^{k}_{i},\bar{y}_{-i})\succsim x$ for all $k$ and
$x\succ(y_{i},\bar{y}_{-i})$. By separate convexity, $A_{\succsim}(x)\cap
L_{i}$ is convex. Therefore, if $y^{k}_{i}\leqslant y_{i}\leqslant y^{m}_{i}$
for some $k,m$, then $(y_{i},\bar{y}_{-i})\succsim x$. Therefore, either
$y^{n}_{i}>y_{i}$ for all $n$ or $y^{n}_{i}<y_{i}$ for all $n$. Assume wlog
that $y^{n}_{i}>y_{i}$ for all $n$. Then, separate convexity of $\succsim$
implies that $(y^{\prime}_{i},\bar{y}_{-i})\succsim x$ for all
$y^{\prime}_{i}\in(y_{i},y^{1}_{i}]$. It follows from the Archimedean property
and $x\succ(y_{i},\bar{y}_{-i})$ that there exists $\delta\in(0,1)$ such that
$x\succ(y_{i},\bar{y}_{-i})\delta(y_{i}^{1},\bar{y}_{-i})=(y_{i}\delta
y_{i}^{1},\bar{y}_{-i})$. Since $y_{i}\delta y_{i}^{1}\in(y_{i},y^{1}_{i}]$,
this furnishes us a contradiction. Therefore, $\succsim$ is separately upper
continuous. The proof is analogous to the statement that when $\succsim$ has
separately convex lower sections, upper Archimedean implies lower separate
continuity.

For part (b), given the relationships above, it remains to show the
relationship between restricted solvability and separate continuity. In
particular, we show that if $\succsim$ has separately convex lower sections,
then restricted solvability implies that $\succsim$ has separately open lower
sections. Towards this end assume $\succsim$ has separately convex lower
sections,262626This convexity assumption is equivalent to $\succ$ having
separately convex lower sections; see Debreu (1959, p.59) for details and
further equivalences. is restricted solvable but does not have separately open
upper sections. That is, there exists $x\in X$ and a line $L$ parallel to a
coordinate axis such that $A_{\prec}(x)\cap L$ is not open, hence
$A_{\succsim}(x)\cap L$ is not closed. Then, there exist $x\in X$, an index
$i$, and a sequence $y^{k}\rightarrow y$ on the line $L_{i}$ parallel to
coordinate $i$ such that $y^{k}\succsim x$ for all $k$ and $y\prec x$.

Since $y\prec x$, order denseness implies that there exists $x^{\prime}\in X$
such that $y\prec x^{\prime}\prec x$. By restricted solvability, there exists
$z\in L_{i}$ such that $z\sim x^{\prime}$. By transitivity and Sen (1969,
Theorem I), $y\prec z\prec x\precsim y^{k}$ for all $k$. By convexity
assumption and Debreu (1959, p.59), for all $z_{i}^{\prime}\in(y_{i},z_{i}]$,
$(z^{\prime}_{i},z_{-i})\precsim z$, and by transitivity and Sen (1969,
Theorem I), $(z^{\prime}_{i},z_{-i})\prec x$. However, since $y^{k}\rightarrow
y$, there exists $z_{i}^{\prime}\in(y_{i},z_{i}]$, such that
$z_{i}^{\prime}=y_{i}^{k}$ for some $k$ and $(z^{\prime}_{i},z_{-i})\succsim
x$. This yields a contradiction. Therefore, $\succsim$ has closed upper
sections. The proof for closed lower sections of $\succsim$ under separately
convex upper sections is analogous.

Next, we prove the equivalence between separate convexity and separate
monotonicity stated in Section 3.

######  Claim 1.

A complete and transitive binary relation $\succsim$ is separately monotone if
and only if it is monotone if and only if it has separately convex sections.

###### Proof of Claim 1.

We first show that separate monotonicity implies monotonicity. Assume
$\succsim$ is separately monotone (and increasing in each coordinate). Pick
$x>=y$, i.e., $(x_{i}>=y_{i}\text{ for all }i)$. Then, by separate
monotonicity, $(x_{1},y_{-1})\succsim y$. Since
$(x_{1},x_{2},y_{\\{-1,-2\\}})>=(x_{1},y_{-1})$, by separate monotonicity and
transitivity, $(x_{1},x_{2},y_{\\{-1,-2\\}})\succsim y$. By iterating this
argument, we obtain $x\succsim y$, hence $\succsim$ is monotone (increasing).
The proof of the case where $\succsim$ is decreasing in each coordinate is
analogous. The converse relationship is obvious.

To see that separate monotonicity implies separately convex sections, let
$\succsim$ be separately monotone (increasing), $(x_{i},z_{-i})\succsim a$,
$(y_{i},z_{-i})\succsim a$ and $x_{i}>y_{i}$. By monotonicity,
$(x_{i},z_{-i})\succsim(y_{i},z_{-i})$, hence for all $w_{i}\in(y_{i},x_{i})$,
$(w_{i},z_{-i})\succsim(y_{i},z_{-i})$. By transitivity, for all
$w_{i}\in(y_{i},x_{i})$, $(w_{i},z_{-i})\succsim a$. Hence, $\succsim$ has
separately convex upper sections. Now let $(x_{i},z_{-i})\precsim a$,
$(y_{i},z_{-i})\precsim a$ and $x_{i}>y_{i}$. By monotonicity,
$(x_{i},z_{-i})\succsim(y_{i},z_{-i})$, hence for all $w_{i}\in(y_{i},x_{i})$,
$(x_{i},z_{-i})\succsim(w_{i},z_{-i})$. By transitivity, for all
$w_{i}\in(y_{i},x_{i})$, $a\succsim(w_{i},z_{-i})$. Hence, $\succsim$ has
separately convex lower sections.

To see that separately convex sections imply separate monotonicity, assume
$\succsim$ has separately convex sections, pick $i$, $z\in X$, and
$(a_{i},z_{-i}),(b_{i},z_{-i})\in X$. Assume wlog that $a_{i}>b_{i}$ and
$(a_{i},z_{-i})\succsim(b_{i},z_{-i})$. By convexity, for all
$w_{i}\in(b_{i},a_{i})$,
$(a_{i},z_{-i})\succsim(w_{i},z_{-i})\succsim(b_{i},z_{-i})$. Pick
$x_{i}>a_{i}$. Let $(x_{i},z_{-i})\prec(a_{i},z_{-i})$. Assume wlog that
$(x_{i},z_{-i})\succsim(b_{i},z_{-i})$. Then $\succsim$ having convex lower
sections imply that for all $w_{i}\in(b_{i},x_{i})$,
$(x_{i},z_{-i})\succsim(w_{i},z_{-i})$. By transitivity and
$a_{i}\in(b_{i},x_{i})$, $(a_{i},z_{-i})\succ(a_{i},z_{-i})$, which yields a
contradiction. Hence, for all $x_{i}>a_{i}$,
$(x_{i},z_{-i})\succsim(a_{i},z_{-i})$. Analogously, for all $x_{i}<a_{i}$,
$(a_{i},z_{-i})\succsim(x_{i},z_{-i})$. Therefore, $\succsim$ is separately
monotone.

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