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\begin{document}

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\section*{Top}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{UniversalConstructions}{Universal constructions}\dotfill \pageref*{UniversalConstructions} \linebreak
\noindent\hyperlink{relation_with_}{Relation with $Set$}\dotfill \pageref*{relation_with_} \linebreak
\noindent\hyperlink{MonoEpiMorphisms}{Mono-/Epimorphisms}\dotfill \pageref*{MonoEpiMorphisms} \linebreak
\noindent\hyperlink{intersections_and_quotients}{Intersections and quotients}\dotfill \pageref*{intersections_and_quotients} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\textbf{Top} denotes the [[category]] whose [[objects]] are [[topological spaces]] and whose [[morphisms]] are [[continuous functions]] between them. Its [[isomorphisms]] are the [[homeomorphisms]].

For exposition see \emph{[[Introduction to Topology -- 1|Introduction to point-set topology]]}.

Often one considers (sometimes by default) [[subcategories]] of [[nice topological spaces]] such as [[compactly generated topological spaces]], notably because these are [[cartesian closed category|cartesian closed]]. There other other [[convenient categories of topological spaces]]. With any one such choice understood, it is often useful to regard it as ``the'' category of topological spaces.

The [[homotopy category]] of $Top$ given by its [[localization]] at the [[weak homotopy equivalences]] is the [[classical homotopy category]] [[Ho(Top)]]. This is the central object of study in [[homotopy theory]], see also at \emph{[[classical model structure on topological spaces]]}. The [[simplicial localization]] of [[Top]] at the [[weak homotopy equivalences]] is the archetypical [[(∞,1)-category]], [[equivalence of (infinity,1)-categories|equivalent]] to [[∞Grpd]] (see at \emph{[[homotopy hypothesis]]}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{UniversalConstructions}{}\subsubsection*{{Universal constructions}}\label{UniversalConstructions}

We discuss [[universal constructions]] in [[Top]], such as [[limits]]/[[colimits]], etc. The following definition suggests that universal constructions be seen in the context of $Top$ as a [[topological concrete category]] (see Proposition \ref{topcat} below).

$\,$

[[!include universal constructions of topological spaces -- table]]

$\,$

\begin{defn}
\label{InitialAndFinalTopologies}\hypertarget{InitialAndFinalTopologies}{}
Let $\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}$ be a [[class]] of [[topological spaces]], and let $S \in Set$ be a bare [[set]]. Then

\begin{itemize}%
\item For $\{S \stackrel{f_i}{\to} S_i \}_{i \in I}$ a set of [[functions]] out of $S$, the \emph{[[initial topology]]} $\tau_{initial}(\{f_i\}_{i \in I})$ is the topology on $S$ with the [[minimum]] collection of [[open subsets]] such that all $f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i$ are [[continuous function|continuous]].


\item For $\{S_i \stackrel{f_i}{\to} S\}_{i \in I}$ a set of [[functions]] into $S$, the \emph{[[final topology]]} $\tau_{final}(\{f_i\}_{i \in I})$ is the topology on $S$ with the [[maximum]] collection of [[open subsets]] such that all $f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))$ are [[continuous function|continuous]].



\end{itemize}
\end{defn}
\begin{example}
\label{TopologicalSubspace}\hypertarget{TopologicalSubspace}{}
For $X$ a single topological space, and $\iota_S \colon S \hookrightarrow U(X)$ a subset of its underlying set, then the initial topology $\tau_{intial}(\iota_S)$, def. \ref{InitialAndFinalTopologies}, is the [[subspace topology]], making

\begin{displaymath}
\iota_S
  \;\colon\;
  (S, \tau_{initial}(\iota_S))
  \hookrightarrow
  X
\end{displaymath}
a [[topological subspace]] inclusion.

\end{example}
\begin{example}
\label{QuotientTopology}\hypertarget{QuotientTopology}{}
Conversely, for $p_S \colon U(X) \longrightarrow S$ an [[epimorphism]], then the final topology $\tau_{final}(p_S)$ on $S$ is the \emph{[[quotient topology]]}.

\end{example}
\begin{prop}
\label{DescriptionOfLimitsAndColimitsInTop}\hypertarget{DescriptionOfLimitsAndColimitsInTop}{}
Let $I$ be a [[small category]] and let $X_\bullet \colon I \longrightarrow Top$ be an $I$-[[diagram]] in [[Top]] (a [[functor]] from $I$ to $Top$), with components denoted $X_i = (S_i, \tau_i)$, where $S_i \in Set$ and $\tau_i$ a topology on $S_i$. Then:

\begin{enumerate}%
\item The [[limit]] of $X_\bullet$ exists and is given by [[generalized the|the]] topological space whose underlying set is [[generalized the|the]] limit in [[Set]] of the underlying sets in the diagram, and whose topology is the [[initial topology]], def. \ref{InitialAndFinalTopologies}, for the functions $p_i$ which are the limiting [[cone]] components:

\begin{displaymath}
\itexarray{
    && \underset{\longleftarrow}{\lim}_{i \in I} S_i
    \\
    & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}}
    \\
    S_i && \underset{}{\longrightarrow} && S_j
  }
  \,.
\end{displaymath}
Hence

\begin{displaymath}
\underset{\longleftarrow}{\lim}_{i \in I} X_i  
  \simeq
  \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)
\end{displaymath}

\item The [[colimit]] of $X_\bullet$ exists and is the topological space whose underlying set is the colimit in [[Set]] of the underlying diagram of sets, and whose topology is the [[final topology]], def. \ref{InitialAndFinalTopologies} for the component maps $\iota_i$ of the colimiting [[cocone]]

\begin{displaymath}
\itexarray{
    S_i && \longrightarrow && S_j
    \\
    & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
    \\
    && \underset{\longrightarrow}{\lim}_{i \in I} S_i
  }
  \,.
\end{displaymath}
Hence

\begin{displaymath}
\underset{\longrightarrow}{\lim}_{i \in I} X_i  
  \simeq
  \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right)
\end{displaymath}


\end{enumerate}
\end{prop}
(e.g. \hyperlink{Bourbaki71}{Bourbaki 71, section I.4})

\begin{proof}
The required [[universal property]] of $\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$ is immediate: for

\begin{displaymath}
\itexarray{
    && (S,\tau)
    \\
    & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} 
    \\
    X_i && \underset{}{\longrightarrow} && X_j
  }
\end{displaymath}
any [[cone]] over the diagram, then by construction there is a unique function of underlying sets $S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i$ making the required diagrams commute, and so all that is required is that this unique function is always [[continuous function|continuous]]. But this is precisely what the initial topology ensures.

The case of the colimit is [[formal dual|formally dual]].

\end{proof}
\begin{example}
\label{PointTopologicalSpaceAsEmptyLimit}\hypertarget{PointTopologicalSpaceAsEmptyLimit}{}
The limit over the empty diagram in $Top$ is the [[point]] $\ast$ with its unique topology.

\end{example}
\begin{example}
\label{DisjointUnionOfTopologicalSpacesIsCoproduct}\hypertarget{DisjointUnionOfTopologicalSpacesIsCoproduct}{}
For $\{X_i\}_{i \in I}$ a set of topological spaces, their [[coproduct]] $\underset{i \in I}{\sqcup} X_i \in Top$ is their \emph{[[disjoint union]]}.

\end{example}
In particular:

\begin{example}
\label{DiscreteTopologicalSpaceAsCoproduct}\hypertarget{DiscreteTopologicalSpaceAsCoproduct}{}
For $S \in Set$, the $S$-indexed [[coproduct]] of the point, $\underset{s \in S}{\coprod}\ast$, is the set $S$ itself equipped with the [[final topology]], hence is the [[discrete topological space]] on $S$.

\end{example}
\begin{example}
\label{ProductTopologicalSpace}\hypertarget{ProductTopologicalSpace}{}
For $\{X_i\}_{i \in I}$ a set of topological spaces, their [[product]] $\underset{i \in I}{\prod} X_i \in Top$ is the [[Cartesian product]] of the underlying sets equipped with the \emph{[[product topology]]}, also called the \emph{[[Tychonoff product]]}.

In the case that $S$ is a [[finite set]], such as for binary product spaces $X \times Y$, then a [[basis for a topology|sub-basis]] for the product topology is given by the [[Cartesian products]] of the open subsets of (a basis for) each factor space.

\end{example}
\begin{example}
\label{EqualizerInTop}\hypertarget{EqualizerInTop}{}
The [[equalizer]] of two [[continuous functions]] $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the equalizer of the underlying functions of sets

\begin{displaymath}
eq(f,g) 
   \hookrightarrow 
  S_X 
    \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} 
  S_Y
\end{displaymath}
(hence the largets subset of $S_X$ on which both functions coincide) and equipped with the [[subspace topology]], example \ref{TopologicalSubspace}.

\end{example}
\begin{example}
\label{CoequalizerInTop}\hypertarget{CoequalizerInTop}{}
The [[coequalizer]] of two [[continuous functions]] $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the coequalizer of the underlying functions of sets

\begin{displaymath}
S_X 
    \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} 
  S_Y
    \longrightarrow 
  coeq(f,g)
\end{displaymath}
(hence the [[quotient set]] by the [[equivalence relation]] generated by $f(x) \sim g(x)$ for all $x \in X$) and equipped with the [[quotient topology]], example \ref{QuotientTopology}.

\end{example}
\begin{example}
\label{PushoutInTop}\hypertarget{PushoutInTop}{}
For

\begin{displaymath}
\itexarray{
     A &\overset{g}{\longrightarrow}& Y
     \\
     {}^{\mathllap{f}}\downarrow
     \\
     X
  }
\end{displaymath}
two [[continuous functions]] out of the same [[domain]], then the [[colimit]] under this diagram is also called the \emph{[[pushout]]}, denoted

\begin{displaymath}
\itexarray{
     A &\overset{g}{\longrightarrow}& Y
     \\
     {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}}
     \\
     X &\longrightarrow& X \sqcup_A Y
     \,.
  }
  \,.
\end{displaymath}
(Here $g_\ast f$ is also called the pushout of $f$, or the \emph{[[base change|cobase change]]} of $f$ along $g$.) If $g$ is an inclusion, one also write $X \cup_f Y$ and calls this the \emph{[[attaching space]]}.

By example \ref{CoequalizerInTop} the [[pushout]]/[[attaching space]] is the [[quotient topological space]]

\begin{displaymath}
X \sqcup_A Y \simeq (X\sqcup Y)/\sim
\end{displaymath}
of the [[disjoint union]] of $X$ and $Y$ subject to the [[equivalence relation]] which identifies a point in $X$ with a point in $Y$ if they have the same pre-image in $A$.

(graphics from \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02})

\end{example}
\begin{example}
\label{TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself}\hypertarget{TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself}{}
As an important special case of example \ref{PushoutInTop}, let

\begin{displaymath}
i_n \colon S^{n-1}\longrightarrow D^n
\end{displaymath}
be the canonical inclusion of the standard [[n-sphere|(n-1)-sphere]] as the [[boundary]] of the standard [[n-disk]] (both regarded as [[topological spaces]] with their [[subspace topology]] as subspaces of the [[Cartesian space]] $\mathbb{R}^n$).

Then the colimit in [[Top]] under the diagram, i.e. the [[pushout]] of $i_n$ along itself,

\begin{displaymath}
\left\{
     D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow}
     D^n
  \right\}
  \,,
\end{displaymath}
is the [[n-sphere]] $S^n$:

\begin{displaymath}
\itexarray{
    S^{n-1} &\overset{i_n}{\longrightarrow}& D^n
    \\
    {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow
    \\
    D^n &\longrightarrow& S^n
  }
  \,.
\end{displaymath}
(graphics from Ueno-Shiga-Morita 95)

\end{example}
\begin{example}
\label{ClosedSubspacesGluing}\hypertarget{ClosedSubspacesGluing}{}
\textbf{([[union]] of two [[open subset|open]] or two [[closed subset|closed]] [[subspaces]] is [[pushout]])}

Let $X$ be a [[topological space]] and let $A,B \subset X$ be [[subspaces]] such that

\begin{enumerate}%
\item $A,B \subset X$ are both [[open subsets]] or are both [[closed subsets]];


\item they constitute a [[cover]]: $X = A \cup B$



\end{enumerate}
Write $i_A \colon A \to X$ and $i_B \colon B \to X$ for the corresponding inclusion [[continuous functions]].

Then the [[commuting square]]

\begin{displaymath}
\itexarray{
     A \cap B &\longrightarrow& A
     \\
     \downarrow && \downarrow^{\mathrlap{i_A}}
     \\
     B &\underset{i_B}{\longrightarrow}& X
  }
\end{displaymath}
is a [[pushout]] square in $Top$ (example \ref{PushoutInTop}).

By the [[universal property]] of the [[pushout]] this means in particular that for $Y$ any [[topological space]] then a function of underlying sets

\begin{displaymath}
f \;\colon\; X \longrightarrow Y
\end{displaymath}
is a [[continuous function]] as soon as its two restrictions

\begin{displaymath}
f\vert_A \;\colon\; A \longrightarrow Y
  \phantom{AAAA}
  f\vert_A \;\colon\; B \longrightarrow Y
\end{displaymath}
are continuous.

\end{example}
\begin{proof}
Clearly the underlying diagram of underlying [[sets]] is a pushout in [[Set]]. Therefore by prop. \ref{DescriptionOfLimitsAndColimitsInTop} we need to show that the [[topological space|topology]] on $X$ is the [[final topology]] induced by the set of functions $\{i_A, i_B\}$, hence that a [[subset]] $S \subset X$ is an [[open subset]] precisely if the [[pre-images]] (restrictions)

\begin{displaymath}
i_A^{-1}(S) = S \cap A
   \phantom{AAA}
    \text{and}
   \phantom{AAA}
  i_B^{-1}(S) = S \cap B
\end{displaymath}
are open subsets of $A$ and $B$, respectively.

Now by definition of the [[subspace topology]], if $S \subset X$ is open, then the intersections $A \cap S \subset A$ and $B \cap S \subset B$ are open in these subspaces.

Conversely, assume that $A \cap S \subset A$ and $B \cap S \subset B$ are open. We need to show that then $S \subset X$ is open.

Consider now first the case that $A;B \subset X$ are both open open. Then by the nature of the [[subspace topology]], that $A \cap S$ is open in $A$ means that there is an open subset $S_A \subset X$ such that $A \cap S = A \cap S_A$. Since the intersection of two open subsets is open, this implies that $A \cap S_A$ and hence $A \cap S$ is open. Similarly $B \cap S$. Therefore

\begin{displaymath}
\begin{aligned}
    S 
     & = 
    S \cap X 
    \\
    & = S \cap (A \cup B) 
    \\
    & =  (S \cap A) \cup (S \cap B)
  \end{aligned}
\end{displaymath}
is the union of two open subsets and therefore open.

Now consider the case that $A,B \subset X$ are both closed subsets.

Again by the nature of the subspace topology, that $A \cap S \subset A$ and $B \cap S \subset B$ are open means that there exist open subsets $S_A, S_B \subset X$ such that $A \cap S = A \cap S_A$ and $B \cap S = B \cap S_B$. Since $A,B \subset X$ are closed by assumption, this means that $A \setminus S, B \setminus S \subset X$ are still closed, hence that $X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X$ are open.

Now observe that (by [[de Morgan duality]])

\begin{displaymath}
\begin{aligned}
    S 
      & = X \setminus (X \setminus S)
    \\
      & = 
    X \setminus ( (A \cup B) \setminus S )
    \\
      & =
    X \setminus ( (A \setminus S) \cup (B \setminus S) )
    \\
     & =
    (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S))
    \,.
  \end{aligned}
\end{displaymath}
This exhibits $S$ as the intersection of two open subsets, hence as open.

\end{proof}
\begin{example}
\label{attach}\hypertarget{attach}{}
If $X, Y, Z$ are [[normal topological spaces]] and $h: X \to Z$ is a [[closed embedding of topological spaces]] and $f: X \to Y$ is a [[continuous function]], then in the [[pushout]] diagram in $Top$ (example \ref{PushoutInTop})

\begin{displaymath}
\itexarray{
X & \stackrel{h}{\to} & Z \\ 
\mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\ 
Y & \underset{k}{\to} & W,
}
\end{displaymath}
the space $W$ is normal and $k: Y \to W$ is a closed embedding.

\end{example}
For \textbf{proof} of this and related statements see at \emph{[[colimits of normal spaces]]}.

\hypertarget{relation_with_}{}\subsubsection*{{Relation with $Set$}}\label{relation_with_}

Write [[Set]] for the [[category]] of [[sets]].

\begin{defn}
\label{ForgetfulFunctorFromTopToSet}\hypertarget{ForgetfulFunctorFromTopToSet}{}
Write

\begin{displaymath}
U \colon Top \longrightarrow Set
\end{displaymath}
for the [[forgetful functor]] that sends a topological space $X = (S,\tau)$ to its underlying set $U(X) = S \in Set$ and which regards a [[continuous function]] as a plain [[function]] on the underlying sets.

\end{defn}
Prop. \ref{DescriptionOfLimitsAndColimitsInTop} means in particular that:

\begin{prop}
\label{}\hypertarget{}{}
The category [[Top]] has all small [[limits]] and [[colimits]]. The [[forgetful functor]] $U \colon Top \to Set$ from def. \ref{ForgetfulFunctorFromTopToSet} [[preserved limit|preserves]] and [[lifted limit|lifts]] limits and colimits.

\end{prop}
(But it does not [[created limit|create]] or [[reflected limit|reflect]] them.)

\begin{prop}
\label{}\hypertarget{}{}
The [[forgetful functor]] $U$ from def. \ref{ForgetfulFunctorFromTopToSet} has a [[left adjoint]] $disc$, given by sending a [[set]] $S$ to the corresponding [[discrete topological space]], example \ref{DiscreteTopologicalSpaceAsCoproduct}

\begin{displaymath}
Top
  \stackrel{\overset{disc}{\longleftarrow}}{\underset{U}{\longrightarrow}}
  Set
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{topcat}\hypertarget{topcat}{}
The [[forgetful functor]] $U$ from def. \ref{ForgetfulFunctorFromTopToSet} exhibits $Top$ as

\begin{itemize}%
\item a [[concrete category]]


\item a [[topological concrete category]].



\end{itemize}
\end{prop}
\hypertarget{MonoEpiMorphisms}{}\subsubsection*{{Mono-/Epimorphisms}}\label{MonoEpiMorphisms}

\begin{prop}
\label{SubspaceInclusionsAreRegularMonos}\hypertarget{SubspaceInclusionsAreRegularMonos}{}
\textbf{([[regular monomorphisms]] of [[topological spaces]])}

In the [[category]] [[Top]] of [[topological space]],

\begin{enumerate}%
\item the [[monomorphisms]] are the those [[continuous functions]] which are [[injective functions]];


\item the [[regular monomorphisms]] are the [[topological embeddings]] (i.e. those continuous functions which are [[homeomorphisms]] onto their [[images]] equipped with the [[subspace topology]]).



\end{enumerate}
\end{prop}
\begin{proof}
Regarding the first statement: An injective continuous function $f \colon X \to Y$ clearly has the cancellation property that defines monomorphisms: for parallel continuous functions $g_1,g_2 \colon Z \to X$: if $f \circ g_1 = f \circ g_1$, then $g_1 = g_2$ because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if $f$ has the cancellation property, then testing on points $g_1, g_2 \colon \ast \to X$ gives that $f$ is injective.

Regarding the second statement: from the construction of [[equalizers]] in [[Top]] (example \ref{EqualizerInTop}) we have that these are topological subspace inclusions.

Conversely, let $i \colon X \to Y$ be a [[topological subspace embedding]]. We need to show that this is the equalizer of some pair of parallel morphisms.

To that end, form the [[cokernel pair]] $(i_1, i_2)$ by taking the [[pushout]] of $i$ against itself (in the category of sets, and using the [[quotient topology]] on a [[disjoint union space]]). By \href{regular+monomorphism#RegEquEff}{this prop.}, the equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the [[subspace topology]]. Since monomorphisms in [[Set]] are regular, we get the function $i$ back, and again by example \ref{EqualizerInTop}, it gets equipped with the subspace topology. This completes the proof.

\end{proof}
\hypertarget{intersections_and_quotients}{}\subsubsection*{{Intersections and quotients}}\label{intersections_and_quotients}

\begin{lemma}
\label{pushout}\hypertarget{pushout}{}
The [[pushout]] in [[Top]] of any (closed/open) [[topological subspace]] inclusion $i \colon A \hookrightarrow B$, example \ref{TopologicalSubspace}, along any [[continuous function]] $f \colon A \to C$ is itself an a (closed/open) subspace $j \colon C \hookrightarrow D$.

\end{lemma}
For proof see \href{subspace+topology#pushout}{there}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[topological concrete category]]


\item [[Ho(Top)]]


\item [[convenient category of topological spaces]]


\item [[TopGrp]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For general references see those listed at \emph{[[topology]]}, such as

\begin{itemize}%
\item [[Nicolas Bourbaki]], chapter 1 \emph{Topological Structures} of \emph{Elements of Mathematics III: General topology}, Springer 1971, 1990

\end{itemize}
See also

\begin{itemize}%
\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})

\end{itemize}
An axiomatic desciption of $Top$ along the lines of [[ETCS]] for [[Set]] is discussed in

\begin{itemize}%
\item Dana Schlomiuk, \emph{An elementary theory of the category of topological space}, Transactions of the AMS, volume 149 (1970)

\end{itemize}
category: category

[[!redirects category of topological spaces]]





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