$\Sier_B$ is a suitable subcategory of $\Top_B$. By Theorem <ref>,	$\Sier_B$ is left Kan extendable.

The subcategory $\Sier$ of $\Top$ corresponds to the subcategory $\Sier_{\Pt}$ of $\Top_{\Pt}$ under the isomorphism $P$ of (<ref>).
We therefore have the following.

$\Sier$ is  left Kan extendable subcategory in $\Top$.

A fibrewise topological space $X \ra B$ is $\Sier_B$-generated iff its domain $X$ is $\Sier$-generated.

This is a consequence  of Proposition <ref>.1  and lemma <ref>.1.

Recall that an Alexandroff space is a topological space in which arbitrary  intersections of open subsets are open. Equivalently,
an Alexandroff space is a topological space for which arbitrary  unions of closed subsets are closed. Let $\Alex$ be the subcategory of $\Top$ of Alexandroff spaces. A finite topological space has only finitely many open sets, and is therefore an Alexandroff space.

Let $B$ be a subset of an Alexandroff space $X$ and let $\overline{B}$ denote the topological closure of $B$. The subspace $\underset{b\in B}{\bigcup}\overline{\{b\}}$ is a closed subset of $X$ containing
$B$. It follows that $\overline{B}=\underset{b\in B}{\bigcup}\overline{\{b\}}$.

We next provide a simple proof of the following result which is given (without proof) in <cit.>.

A topological space $X$ is $\Sier$-generated iff it is an Alexandroff space. That is $\Sier_{l}[\Top]=\Alex$.

Let $X$ be a $\Sier$-generated topological space, $(O_i)_{i\in I}$ a family of open sets in $X$ and $f:\bs \ra X$ a continuous map. Then
$f^{-1}(\underset{i\in I}{\bigcap} O_i)=\underset{i\in I}{\bigcap}f^{-1}(O_i)$ which open in $\bs$ since $\bs$ is an Alexandroff space. By Corollary <ref>.2, $\underset{i\in I}{\bigcap} O_i$ is open in $X$ and $X$ is an Alexandroff space.
Conversely, assume that $X$ is an Alexandroff space. Let $B \subset X$ be such that $f^{-1}(B)$ is closed for every continuous map $f:\bs \ra X$  and let $a\in \overline{B}$. There exists $b\in B$ such that $a\in \overline{\{b\}}$. If $a=b$ then $a\in B$, if $a\neq b$, define $g:\bs \ra X$ by $g(0)=a$ and $g(1)=b$. Then

$$g(\overline{\{1\}})=g(\bs)=\{a,b\}\subset \overline{\{b\}} \subset \overline{g(\{1\})}$$

Therefore $g$ is continuous and $g^{-1}(B)$ is a closed subset of $\bs$ containing $1$. It follows that $g^{-1}(B)=\bs$, in particular, $a=g(0)\in B$ and $B$ is closed in $X$. By Corollary <ref>.2, $X$ is $\Sier$-generated.

It follows that $(\Sier_B)_l[\Top_B]$ is the subcategory $\Alex_B$ of  fibrewise spaces $X \sra B$ whose domain $X$ is an Alexandroff space.
Objects of $\Alex_B$ are called fibrewise Alexandroff spaces over $B$.

* $\Alex$ is a coreflective subcategory of $\Top $ containing $\Sier$ as a dense subcategory.

* $\Alex_B$ is a coreflective subcategory of $\Top_B $ containing $\Sier_B$ as a dense subcategory.

This follows from Proposition <ref>, Proposition <ref>,  Proposition <ref>.1  and Proposition <ref>.

We next generalize   <cit.>.

* The Sierpinski space $\bs$ is sequential.

* The category $\Alex$ is a coreflective subcategory of $\Seq$.

* The category $\Alex_B$ is a coreflective subcategory of $\Seq_B$.

Define $q:\bn^{+} \ra  \bs$ by $q(\infty)=0$ and $q(n)=1$ for all $n\in \bn$. The map $q$ is a quotient map, thus  by Proposition <ref>.1, $\bs$ is sequential. By Corollary <ref>, $\Alex$ is a coreflective subcategory of $\Seq$.  Similarly, $\Alex_B$ is a coreflective subcategory of $\Seq_B$.

* The subcategory $\Alex$ of $\Top$ is cartesian closed.

* If $B$ is $\T_1$, then the subcategory $\Alex_B$ of $\Top_B$ is cartesian closed.

We just need to prove 2.

Being finite, $\bs$ is a core-compact space. It is therefore an exponentiable object of $\Top$. Let $\bs \st{p}\ra B$ be continuous.
Assume the space $B$ is $\T_1$, therefore $p$ is constant. By	Proposition <ref>, $\bs \st{p}\ra B$ is an exponentiable object of $\Top_B$. The product in $\Top_B$ of two fibrewise Sierpinski spaces is a fibrewise Alexandroff space.  By Theorem <ref>,
$\Alex_B$ is cartesian closed.

* Assume that $B$ is an Alexandroff space. Then $\Alex_B$ is just the slice category $\Alex/B$ and the adjunction given by Lemma <ref>.2 yields an adjunction

\begin{equation}\label{eq54}
\begin{tikzcd}
\Alex_B\arrow[r, shift left=1ex, "P_B"{name=G}] &  \Alex\arrow[l, shift left=.3ex, ".\tm_{\Alex}B" ].
\end{tikzcd}
\end{equation}

* Let $a:\Top \ra \Alex$ be a coreflector. Then the functor
$\Alex_B \ra \Alex_{a(B)}$ which takes a fibrewise Alexandroff space $X \st{p}\ra B$ to $ X\st{a(p)}\ra a(B)$ is an isomorphism of categories.

* By the previous two points, for any topological space $B$, the functor $P_B:\Alex_B \ra \Alex$
is left adjoint. Its
right adjoint	takes  an Alexandroff space $X$ to the fibrewise Alexandroff space $X\tm_{\Alex}a(B)$ with projection the composite
$$X\tm_{\Alex}a(B) \st{pr}\ra a(B) \st{\epsilon_B}\ra B.$$

Where $a:\Top \ra \Alex$ is a coreflector and $\epsilon_B$ is the $B$-component of the counit $\epsilon$ of the of the coreflection of $\Top$ on  $\Sier$.

§ LIMITS IN A SLICE CATEGORY

The aim of this section is to prove that if $\mc$ is a bicomplete category and $b\in \mc$, then  the slice category $\mc/b$ of $\mc$  over $b$ is bicomplete.

Let $F: \ma \ra \mc$ an $G: \mb \ra \mc$ be two functors. The comma category $F/G$ is defined to be the category whose objects are arrows
$F(a)\st{\alpha}\ra G(b)$ and whose morphisms from $F(a)\st{\alpha}\ra G(b)$
to $F(a')\st{\alpha'}\ra G(b')$ are pairs of morphisms $(f,g)\in \ma(a,a') \tm_{\Set}  \mb(b,b')$ rendering commutative the diagram

\begin{equation}\label{eq21}
\begin{tikzcd}[sep=large]
G(b) \arrow[r,"G(g)"']&G(b')\\
\end{tikzcd}
\end{equation}

We have functors

\begin{equation}\label{eq22}
\begin{matrix}
P: F/G \ra \ma  &\mbox{and} &Q: F/G \ra \mb
\end{matrix}
\end{equation}

defined as follows: if $F(a)\st{\alpha}\ra G(b)\in F/G$, then
$P(\alpha)=a$ and $Q(\alpha)=b$. If $(f,g)$ is a morphism from $F(a)\st{\alpha}\ra G(b)$
to $F(a')\st{\alpha'}\ra G(b')$ as in (<ref>), then $P((f,g))=f$ and $Q((f,g))=g$.

Let $F: \ma \ra \mc$ an $G: \mb \ra \mc$ be two functors.

* If $\ma$ is a subcategory of $\mc$ and $F: \ma \ra \mc$ is the inclusion functor, then $F/G$ is also denoted by $\ma/G$.

* If $\mb$ is a subcategory of $\mc$ and $G: \mb \ra \mc$ is the inclusion functor, then $F/G$ is also denoted by $F/\mb$.

* If $\ma,\mb$ are  subcategories of $\mc$ and $F: \ma \ra \mc$, $G: \mb \ra \mc$ are the inclusion functors, then $F/G$ is denoted by $\ma/\mb$.

* If $\ma$ has just one object $*$ and just one morphisms $id_*$, then $F/G$ is denoted by $c/G$ where $c=F(*)$. If further $\mb=\mc$ and $G$ is the identity functor, then $c/G$ is called the slice category under $c$ and is denoted by $c/\mc$.

* If $\mb$ has just one object $*$ and just one morphisms $id_*$, then $F/G$ is denoted by $F/c$ where $c=G(*)$. If further $\ma=\mc$ and $F$ is the identity functor, then $F/c$ is called the slice category over $c$ and is denoted by $\mc/c$. Observe that this notation is consistent with the one  previously used.

Let $\mc$ be category, $b\in \mc$, $\mc/b$ the slice category of $\mc$ over $b$ and define

\begin{equation}\label{eq19}
P_b:   \mc/b \ra \mc
\end{equation}

to be the functor which takes an arrow-object $c\rightarrow b$ to its domain $c$.

* The functor $P_b$ creates colimits. In particular, if $\mc$ is cocomplete, then so is $\mc/b$.

* Assume that the categorical product $c\tm_{\mc} b$ exists for every
$c\in \mc$, then $P_b$ is left adjoint. In particular $P_b$ preserves colimits.

* This follows from the dual of a straightforward generalization of  <cit.>.

* The functor $\mc \ra \mc/b$ which takes an object $c\in \mc$ to the arrow $c\tm_{\mc} b \rightarrow b$ is a right adjoint of $P_b$. Thus $P_b$ preserves colimits.

Let $\Cat$ be the category of small categories. A poset carries a category structure in the standard way. Thus the ordinal numbers $1=\{0\}$ and $2=\{0,1\}$ may be viewed as small categories. The small category $1$ is a terminal object in $\Cat$.
The cone $\mi^{\triangleright}$ of $\mi\in \Cat$ is defined in <cit.>  to be the pushout  in $\Cat$:

\arrow[dr, phantom, "\ulcorner", very near end]
\mi\arrow[r,"*"]&1\arrow[d]\\
\mi\tm_{\Cat} 2 \arrow[hookleftarrow,u,"i_1"]\arrow[r]&\mi^{\triangleright}
\end{tikzcd}$$

Let $i$ be the composite functor  $  \mi \st{i_0}\hra \mi\tm_{\Cat} 2 \rightarrow \mi^{\triangleright}$. Then $i:\mi \hra \mi^{\triangleright}$ is fully faithful and $\mi$ may be viewed as a full subcategory of $\mi^{\triangleright}$. Furthermore

* The category $\mi^{\triangleright}$ contains one more object than $\mi$, it is denoted by $*$.

* The set $\mi^{\triangleright}(i,*)$ contains precisely one morphism denoted by $\sigma_i$, $\forall i \in \mi$.

* The set $\mi^{\triangleright}(*,*)$ contains solely the identity morphism.

* The set $\mi^{\triangleright}(*,i)$ is empty,  $\forall i \in \mi$.

Let $X: \mi \ra \mc/b$ be any functor. Define $X^{\triangleright}: \mi^{\triangleright} \ra \mc$ to be the unique functor satisfying the following properties:

* The functor $X^{\triangleright}$ extends $P_bX$ over the category $\mi^{\triangleright}$. That is the following diagram commutes

\mi\arrow[r,"X"]&\mc/b\arrow[d,"P_b"]\\
\mi^{\triangleright}\arrow[hookleftarrow,u,"i"]\arrow[r,"X^{\triangleright} "']&\mc\\
\end{tikzcd}

* $X^{\triangleright}(*)=b$.

* $X^{\triangleright}(\sigma_i)$ is the arrow $X(i)$ in $\mc$, $ i \in \mi$.

Then one has the following result.

* The functor $X$ has a limit if and only if $X^{\triangleright}$ has a limit. Furthermore, a limiting cone $l\st{\lambda}\Ra X^{\triangleright}$
induces a limiting cone from  $\lambda_*:l\ra b$ to $X$, where
$\lambda_*$ is the $*$-component of the cone $\lambda$.

* If $\mc$ is complete, then so is $\mc/b$.

Assume that $\mc$ is complete

* Let $x\st{\sigma}\ra b, y\st{\tau}\ra b \in \mc/b$, $p_1:x\tm_{\mc} y \ra x$, $p_2:x\tm_{\mc} y \ra y$ be the projections  and let

e\ar[r,"i"]&x\tm_{\mc} y \ar[r,shift left=.5ex,"\sigma p_1"]
\ar[r,shift right=.5ex,swap,"\tau p_2"]
\end{tikzcd}$$

be the equalizer (in $\mc$) of the maps $\sigma p_1$	and $\tau p_2$.
The diagram

e [r, "p_2i"] [d, "p_1i"']
[dr, phantom, "1.5$\lrcorner$" , very near start, color=black]
y [d, "τ"]

x [r, "σ"']     b

is a pullback diagram.
By Lemma <ref>, the composite $\sigma p_1 i=\tau p_2 i:e\ra b$ is the product of the objects
$\sigma$ and $\tau$ of $\mc/b$.

* Using generalized equalizers, the previous example may be extended to the case where one has a family of arrow-objects $x_i\st{\sigma_i}\ra b$ of $\mc/b$ indexed be a small set $I$.

§ LIMITS IN A SLICE CATEGORY OF SETS

The aim of this section is to establish certain properties of limits and colimits in a slice   category of  sets.

The category $\Set$ of (small) sets is a bicomplete category. For $X,Y,Z\in \Set$, there is a natural isomorphism

\begin{equation}\label{eq31}
\Set(X\tm_{\Set}Y,Z)\cong \Set(X,\Set(Y,Z))
\end{equation}

so that $\Set$ is cartesian closed.

Let  $E\in \Set$. The slice category of $\Set$ over $E$ is denoted by $\Set_E$. An object of $\Set_E$ is called a set over $E$. It consists of a set $X$ together with a function $p:X\ra E$ called  projection. A set $X\st{p}\ra E$ over $E$ is often identified with its domain $X$. Let

\begin{equation}\label{eq23}
P_E:   \Set_E \ra \Set
\end{equation}

be the functor defined as in (<ref>).

* The category $\Set_E$ is bicomplete.

* The functor $P_E$ creates and preserves colimits.

This follows from Lemma <ref>, Lemma <ref> and the fact that $\Set$ is bicomplete.

Let $F\subset E$, $J^{s}_{F}:F \sra E$ the inclusion map and

\begin{equation}\label{eq29}
J^{s,*}_{F}:\Set_E \sra \Set_F
\end{equation}

the functor given by pulling back along the inclusion map $J^{s}_{F}$.

Let $F\subset E$. Then the functor
$J^{s,*}_{F}: \Set_E \ra \Set_F$ preserves both limits and colimits.

Clearly, $J^{s,*}_{F}$ is at once a right adjoint and  a left adjoint. Therefore it preserves both limits and colimits.

Let $e\in E$. For $X\in \Set_E$, define the fibre of $X$ over $e$ to be the set $X_e=p^{-1}(e)$, where $p$ is the projection of the  set $X$ over $E$. One has a functor

\begin{equation}\label{eq27}
\pi^{s}_{e}:\Set_E \ra \Set
\end{equation}

defined as follows: For $X\in \Set_E$, $\pi^{s}_{e}(X)=X_e$ and for $f\in \Set_E(X,Y)$, $\pi^{s}_{e}(f)$ is the map $f_e:X_e\ra Y_e$ induced by $f$.

* The functors $\pi^{s}_{e}$, $e\in E$, preserve limits and a functor $X:\mi\ra \Set_E$ has a limit iff the composite functor
$\pi^{s}_{e}X$ has a limit for all $e\in E$.

* The functors $\pi^{s}_{e}$, $e\in E$, preserve colimits and a functor  $X:\mi\ra \Set_E$ has a colimit iff the composite functor
$\pi^{s}_{e}X$ has a colimit for all $e\in E$.

By  Lemma <ref>,  $\pi^{s}_{e}$ preserves limits and colimits. The other properties are easy to verify.

Let $X,Y\in \Set_E$.  Then

\begin{equation}\label{eq34}
P_E(X)\cong\underset{\Set}{\overset{e\in E}{\coprod}} X_e
\end{equation}

and a map $f:X\ra Y$ can be written as

\begin{equation}\label{eq35}
f= \underset{\Set}{\overset{e\in E}{\coprod}}f_e: \underset{\Set}{\overset{e\in E}{\coprod}} X_e\ra  \underset{\Set}{\overset{e\in E}{\coprod}} Y_e
\end{equation}

so that one has a natural isomorphism

\begin{equation}\label{eq32}
\Set_E(X,Y)\cong\Set_E( \underset{\Set}{\overset{e\in E}{\coprod}} X_e, \underset{\Set}{\overset{e\in E}{\coprod}} Y_e)\cong  \underset{\Set}{\overset{e\in E}{\prod}}\Set(X_e,Y_e)
\end{equation}

\begin{equation}\label{eq33}
\begin{array}{rrcl}
(.)^{(.)}:&\Set^{op}_E\tm \Set_E&\ra &\Set_E\\
&(Y,Z) &\mapsto &Z^Y= \underset{\Set}{\overset{e\in E}{\coprod}}\Set(Y_e,Z_e)
\end{array}
\end{equation}

That is, $Z^Y$ is the set over $E$ whose fibre over $e \in E $ is $\Set(Y_e,Z_e)$.

Let $X,Y,Z\in \Set_E$. By Lemma <ref>.1,
$(X\tm_{\Set_E} Y)_e=X_e\tm_{\Set} Y_e$. Therefore

$$ \begin{array}{rclr}
\Set_E(X\tm_{\Set_E} Y,Z)&\cong& \underset{\Set}{\overset{e\in E}{\prod}}\Set(X_e\tm_{\Set} Y_e,Z_e)    & (\mbox{by } (\ref{eq32}))\\
&\cong& \underset{\Set}{\overset{e\in E}{\prod}}\Set(X_e,\Set(Y_e,Z_e))  &(\mbox{by } (\ref{eq31}))\\
&\cong&\Set_E(X,Z^Y) & (\mbox{by (\ref{eq32}) and  (\ref{eq33}}))
\end{array}$$

It follows that $\Set_E$ is cartesian closed.

(Category of elements, <cit.> and <cit.>)

* Let $\Set$ be the category of (small) sets and $T:\mi \ra \Set$ be a functor. Recall that

* The category $\int T$ of elements of $T$ is the
category whose objects are pairs $(i,s)$ where $i\in \mi$ and $s\in T(i)$ and morphisms from $(i,s)$ to $(j,t)$ are morphisms
$f$ from $i$ to $j$ satisfying $T(f)(s)=t$.

* The functor $T$ has a colimit iff the  connected components of the category $\int T$ form an object $\pi_0(\int T)\in \Set$, i.e. a  small set . In this case, the cone  $T\st{\lambda}\Ra\pi_0(\int T)$ whose $i$-component is the map

\lambda_i:&T(i) &\ra &\pi_0(\int T)\\
&t &\mapsto &\left[(i,t) \right]
\end{array}$$

is a colimiting cone.

* It follows from the previous point that if a cone $T\st{\lambda}\Ra S$  from the functor $T$ to $S\in \Set$ is such that:

* $\forall s\in T(i)$,   $\forall t\in T(j)$,   $\lambda_i(s)=\lambda_j(t) \Leftrightarrow$ the objects $(i,s)$ and $(j,t)$ of $\int T$ are in the same connected component.

* $ \bigcup_{i\in \mi}  \lambda_i(T(i))=S.$

Then $\lambda$ is a colimiting cone.

* Assume now that $T: I \ra \Set_E$ is a functor. Let  $p_i:T(i) \ra E$ be the projection of the  set $T(i)$ over $E$
$P_E: \Set_E \ra \Set$ the functor given by (<ref>).
Then by  Lemma <ref> and Remark <ref>.1.(b), $T$ has a colimit iff the  connected components of the category $\int P_ET$ form a (small) set. When this is the case, then the colimit of $T$ is

\pi_0(\int P_ET) &\ra &E\\
\left[(i,t) \right] &\mapsto & p_i(t)
\end{array}$$

Let $X:\mi\ra \Set$ be a functor and $Y\in \Set$. Assume that $X\st{\lambda}\Ra Y$ is a colimiting cone and let $Y'\subset Y$. Then the functor $X$ induces a functor $X':\mi\ra \Set$ given by  $X'(i)=\lambda_i^{-1}(Y')$, $i\in \mi$. Furthermore, the cone $\lambda':X'\Ra Y'$   induced by $\lambda$  is a colimiting cone.

Clearly, the functor $X$ induces a functor $X':\mi\ra \Set$
and the cone $\lambda$ induces a cone $\lambda': X'\Ra Y'$. Two objects $(i_1,x_1)$ and $(i_2,x_2)$ in the category $\int X'$ are in the same path-component iff they are in the same path-component of the category $\int X$. That is iff $\lambda'_{i_1}(x_1)=\lambda_{i_1}(x_1)=\lambda_{i_2}(x_2)=\lambda'_{i_2}(x_2)$. Furthermore, $ \bigcup_{i\in \mi}  \lambda'_i(X'(i))=Y'$. By Remark <ref>.3, $\lambda'$ is a colimiting cone.

§ LIMITS IN THE CATEGORY OF FIBREWISE SPACES

Define  $\absl{}:\Top\ra\Set$ to be the underlying set functor. $\abs{.}$  has a left adjoint which is the discrete functor

$$\Disc : \Set \longrightarrow\Top$$

and has a right adjoint which is the codiscrete functor
$$\Codisc :\Set\ra\Top$$
In particular, the underlying functor $\abs{.}$ preserves limits and colimits.

For any functor $T:\mi\ra\Top$, define the underlying set functor of $T$ to be the composite functor

\begin{equation}\label{eq25}
\abs{T}:\mi\st{T}{\ra}\Top\st{\abs{.}}{\ra}\Set.
\end{equation}

Let $\mi$ be a (not necessarily small) category and $T:\mi\ra\Top$ a functor. Then:

* $T$ has a limit (resp. colimit) iff $\abs{T}$ has a limit (resp. colimit).

* Suppose that $\abs{T}$ has a limit (resp. colimit). Then $\lm T$ (resp. $\colim T$) is the topological space whose underlying set is $\lm\abs{T}$ (resp. $\colim \abs{T}$) and whose topology is the initial (resp. final) topology defined by the limiting (resp. colimiting) cone components $\lm\abs{T}\longrightarrow\abs{T(i)}$ (resp. $\abs{T(i)}\longrightarrow\colim\abs{T}$).

It follows from the above lemma that functor $\abs{.}:\Top\ra\Set$ preserves and lifts limits and colimits. $\Set$ is bicomplete, then so is $\Top$.

The slice category of $\Top$ over $B$ is denoted by $\Top_B$. An object of $\Top_B$ is called a fibrewise topological space over $B$. It consists of a topological space $X$ together with a continuous map $p:X\ra B$ called projection. A fibrewise topological space $p:X\ra B$ is often identified with its domain $X$. Let

\begin{equation}\label{eq26}
P_B:   \Top_B \ra \Top
\end{equation}

be the functor defined as in (<ref>).

* $\Top_B$ is bicomplete.

* The functor $P_B$ creates and preserves colimits.

This follows from Lemma <ref> and Lemma <ref>.

If $X$ is a fibrewise space over $B$, then $\abs{X}$ is a set over $\abs{B}$ so that one has an underlying fibrewise set functor

$$\abs{.}:\Top_B\ra\Set_{\abs{B}}. $$

The functor $\abs{.}$ has a left adjoint which is the ordinary discrete functor

$$\Disc : \Set_{\abs{B}} \ra\Top_B,$$

and a right adjoint which is the codiscrete functor
$$\Codisc :\Set_{\abs{B}}\ra\Top_B.$$

It associates to a fibrewise set $S$ over $\abs{B}$ the topological space whose underlying set is $S$ and whose topology is the initial topology defined by the projection $p:S\ra \abs{B}$ of the fibrewise set $S$ on $\abs{B}$.

It follows that the underlying fibrewise set functor $\abs{.}$ preserves both limits and colimits.  We have a commutative diagram of colimit preserving functors

\begin{equation}\label{eq12}
\begin{tikzcd}[sep=large]
\Top_B\ar[r,"\abs{ .}"]\ar[d,"P_B"']&\Set_{\abs{B}}\ar[d,"P_{\abs{B}}"]\\
\Top\ar[r,"\abs{ .}"']&\Set\\
\end{tikzcd}
\end{equation}

where $P_B$ and $P_{\abs{B}}$ are the functors defined by (<ref>) and (<ref>) respectively. For any functor $T:\mi\ra\Top_B$, define the underlying  functor of $T$ to be the composite functor

\begin{equation}\label{eq72}
\abs{T}:\mi\st{T}{\ra}\Top_B\st{\abs{.}}{\ra}\Set_{\abs{B}}
\end{equation}

Let $\mi$ be a (not necessarily small) category and $T:\mi\ra\Top_B$ a functor.

* If one of the functors
$T$, $\abs{T}$, $P_BT$, $\abs{P_BT}=P_{\abs{B}}\abs{T} $
has a colimit, then so do the others.

* Assume that $T$ has a colimit, then $P_B(\colim T)$ is the topological space whose underlying set is $\colim \abs{ P_BT}$
and whose topology is the final topology induced by the components
of the colimiting cone $\abs{P_BT} \Rightarrow \colim \abs{ P_BT}$.

* The functors in diagram (<ref>) are left adjoints, they are therefore colimit preserving, we just need to prove that if $\abs{P_BT}$ has a colimit, then so is $T$. Assume then that $\abs{P_BT}$ has a colimit. By Lemma <ref>, $P_BT$ has a colimit. $P_B$ creates colimits, therefore $T$ has a colimit as desired.

* This follows immediately the first property, Lemma <ref> and the fact that $P_B$ preserves colimits.

Let $\mi$ be a (not necessarily small) category and $T:\mi\ra\Top_B$ a functor. Then:

* $T$ has a limit iff $\abs{T}$ has a limit.

* Assume that $S$ is a  set over $\abs{B}$ and $S \st{\lambda}\Rightarrow \abs{T}$ is a limiting cone. Let $L$ be the topological space whose underlying set is $S$ and whose topology is the initial topology defined by the components of $\lambda$. Then $L$ is a fibrewise space over $B$ and the cone $L\Rightarrow T$ induced by $\lambda$ is a limiting cone.

These are consequences of Lemmas <ref> and <ref>.

Let $X:\mi\ra \Top$ be a functor and $Y\in \Top$. Assume that $X\st{\lambda}\Ra Y$ is a colimiting cone and let $Y'\subset Y$. Let $X':\mi\ra \Top$ be the functor given by  $X'(i)=\lambda_i^{-1}(Y')$, $i\in \mi$. Then the cone $\lambda':X'\Ra Y'$ induced by $\lambda$ is a colimiting cone, provided that $Y'$ is either open or closed in $Y$.

By Lemma <ref>, the cone $\abs{\lambda'}:\abs{X'}\Ra \abs{Y'}$ is a colimiting cone in $\Set$. Assume that $Y'$ is closed, then $X'(i)$ is closed in $X(i)$, all $i\in \mi$. Let $C \subset Y'$ such that
$\lambda'^{-1}_{i}(C)$ is closed in $X'(i)$ for all $i\in I$. The subset $\lambda^{-1}_{i}(C)=\lambda'^{-1}_{i}(C)$ is closed in $X(i)$ for all $i\in I$. By Lemma <ref>, $C$ is closed $Y$ and therefore $C$ is closed $Y'$.
It follows that $Y'$ has the final topology defined by the components of the cone $\abs{\lambda'}$. By Lemma <ref>, $\lambda'$ is a colimiting cone. A similar argument can be used in the case where $Y'$ is open.

Let $A\subset B$, $J^t_A:A \sra B$ the inclusion map and

\begin{equation}\label{eq30}
J^{t,*}_A: \Top_B \ra \Top_A
\end{equation}

The functor given by pulling back along the inclusion map $J^t_A$.

* The functor $J_A^{t,*}$ preserves limits.

* The functor $J_A^{t,*}$ preserves colimits provided that $A$ is either open or closed in $B$.

The first point results from the fact that $J_A^{t,*}$ is a right adjoint. The second is a consequence of Lemma <ref> and Proposition <ref>.

Let $b\in B$. For $X\in \Top_B$, define the fibre of $X$ over $b$ to be the subspace $X_b=p^{-1}(b)$, where $p$ is the projection of the fibrewise space $X$.
One has a functor

\begin{equation}\label{eq28}
\pi^{t}_{b}:\Top_B \ra \Top
\end{equation}

defined as follows: For $X\in \Top_B$, $\pi^{t}_{b}(X)=X_b$ and for $f\in \Top_B(X,Y)$, $\pi^{t}_{b}(f)$ is the map $f_b:X_b\ra Y_b$ is the map induced by $f$.

Let $X:\mi\ra \Top_B$ be a functor.

* The functors $\pi^{t}_{b}$, $b\in B$, preserve limits and  the functor $X$ has a limit iff the composite functor
$\pi^{t}_{b}X$ has a limit for all $b\in B$.

* Assume $B$ is a $\T_1$-space. Then the functors $\pi^{t}_{b}$, $b\in B$, preserve colimits and  the functor $X$ has a colimit iff the composite functor
$\pi^{t}_{b}X$ has a colimit for all $b\in B$.

* By Lemma <ref>.1, the functors $\pi^{t}_{b}$ preserve limits. The fact that $\pi^{t}_{b}X$ has a limit for all $b\in B$
implies that $X$ has a limit is a consequence of   Lemmas <ref>.1, <ref>.1 and <ref>.1.

* By Lemma <ref>.2, the functors $\pi^{t}_{b}$  preserve colimits. The fact that $\pi^{t}_{b}X$ has a colimit for all $b\in B$
implies that $X$ has a colimit is a consequence of  Lemmas <ref>.1, <ref>.2 and <ref>.1.

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