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\begin{document}

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\section*{subspace topology}

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{UniversalProperty}{Universal property}\dotfill \pageref*{UniversalProperty} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \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}

\begin{defn}
\label{SubspaceTopology}\hypertarget{SubspaceTopology}{}
\textbf{([[topological subspace|subspace topology]])}

Let $(X, \tau_X)$ be a [[topological space]], and let $S \subset X$ be a [[subset]] of its underlying [[set]]. Then the corresponding \emph{[[topological subspace]]} has $S$ as its underlying set, and its [[open subsets]] are those subsets of $S$ which arise as restrictions of open subsets of $X$ (i.e. [[intersections]] of open subsets of $X$ with $S$):

\begin{displaymath}
\left(
    U_S \subset S\,\,\text{open}
  \right)
    \,\Leftrightarrow\,
  \left(
     \underset{U_X \in \tau_X}{\exists} \left( U_S = U_X \cap S \right)
  \right)
  \,.
\end{displaymath}
In other words, $\tau_Y$ is the [[finer topology|smallest topology]] on $Y$ such that the inclusion $Y \hookrightarrow X$ is [[continuous map|continuous]] (the [[initial topology]] on that map).

The picture on the right shows two open subsets inside the [[square]], regarded as a [[topological subspace]] of the [[plane]] $\mathbb{R}^2$:

\begin{quote}%
graphics grabbed from \hyperlink{Munkres75}{Munkres 75}


\end{quote}
\end{defn}
The pair $(Y,\tau_Y)$ is then said to be a \emph{topological [[subspace]]} of $(X,\tau_X)$. The induced topology is for that reason sometimes called the \textbf{subspace topology} on $Y$.

A continuous function that factors as a [[homeomorphism]] onto its [[image]] equipped with the subspace topology is called an \emph{[[embedding of topological spaces]]}.

A property of topological spaces is said to be \textbf{hereditary} if its satisfaction for a topological space $X$ implies its satisfaction for all topological subspaces of $X$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The [[open subsets]] of a [[closed interval]] $[0,1] \subset \mathbb{R}$ regarded as a topological subspace of the [[real line]] equipped with its [[Euclidean space|Euclidean]] [[metric topology]] is generated from the [[sub-base of a topology|sub-base]] $\beta = \{ [0, a) ,(a,1]\}_{a \in [0,1]}$.

\end{itemize}
The image on the right shows open subsets in the closed square $[0,1]^2$, regarded as a topological subspace of the Euclidean plane

\begin{itemize}%
\item A subset of a [[metric space]] equipped with its [[induced metric]] is a subspace with respect to the corresponding [[metric topologies]].

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

\hypertarget{UniversalProperty}{}\subsubsection*{{Universal property}}\label{UniversalProperty}

\begin{prop}
\label{UniversalPropertyOfSupspaceInclusion}\hypertarget{UniversalPropertyOfSupspaceInclusion}{}
\textbf{([[universal property]] of [[subspace topology]])}

Let $U \overset{i}{\longrightarrow} X$ be an [[injective function|injective]] [[continuous function]] between [[topological spaces]]. Then this is a [[topological subspace|subspace inclusion]] (Def. \ref{SubspaceTopology}) precisely if it satisfies the following [[universal property]]:

\begin{itemize}%
\item For $Z$ any topological space, a [[function]] $Z \overset{f}{\longrightarrow} U$ (of underlying [[sets]]) is [[continuous function|continuous]] precisely if the [[composition]] $i \circ f$ is continuous as a function to $X$:

\begin{displaymath}
\itexarray{
     Z &\overset{f}{\longrightarrow}& U
     \\
     &{}_{\mathllap{i \circ f}}\searrow& \Big\downarrow{}^{\mathrlap{i}}
     \\
     && X
  }
\end{displaymath}


\end{itemize}
\end{prop}
The elementary \textbf{proof} is spelled out, for instance, in \hyperlink{Terilla14}{Terilla 14, theorem 1}. Of course this is just another way to speak of the [[initial topology]].

The universal characterization of Prop. \ref{UniversalPropertyOfSupspaceInclusion} lends itself to formalization via axioms for [[cohesion]]:

\begin{defn}
\label{SharpModalityOnTopologicalSpaces}\hypertarget{SharpModalityOnTopologicalSpaces}{}
\textbf{([[sharp modality]] on [[topological spaces]])}

Let

\begin{displaymath}
Set
    \underoverset 
      {\underset{coDisc}{\longrightarrow}}
      {\overset{\Gamma}{\longleftarrow}}
      {\bot}
  Top
\end{displaymath}
be the pair of [[adjoint functors]] given by sending a [[topological space]] $X$ to its underlying [[set]] $\Gamma(X)$, and by equipping a [[set]] $S$ with the [[codiscrete topology]] making it a [[codiscrete space]] $coDisc(X)$.

Write

\begin{displaymath}
\sharp \;\coloneqq\; coDisc \circ \Gamma
  \;\colon\;
  Top
    \longrightarrow
  Top
\end{displaymath}
for the induced [[modal operator]] on [[Top]] ([[sharp modality]]). We write

\begin{displaymath}
id \overset{\eta^{\sharp}}{\longrightarrow} \sharp
\end{displaymath}
for the [[unit of an adjunction|unit]] morphism of this [[adjunction]].

Notice that this means that for any [[topological space]] $Z$, \emph{every} function of underlying sets

\begin{displaymath}
Z \longrightarrow \sharp X
\end{displaymath}
is [[continuous functions]], hence that continuous functions into $\sharp X$ are in [[natural bijection]] to underlying functions of [[sets]]. This is the statement of the adjunction hom-isomorphism:

\begin{displaymath}
Hom_{Top}( Z, \sharp X )
  \;\simeq\;
  Hom_{Set}(\Gamma(Z), \Gamma(X))
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{UniversalPropertyViaSharpModality}\hypertarget{UniversalPropertyViaSharpModality}{}
Let $U \overset{i}{\longrightarrow} X$ be an [[injective function|injective]] [[continuous function]] between [[topological spaces]]. Then this is a [[topological subspace|subspace inclusion]] (Def. \ref{SubspaceTopology}) precisely if its [[natural transformation|naturality square]] of the $\sharp$-[[unit of an adjunction|unit]] (Def. \ref{SharpModalityOnTopologicalSpaces})

\begin{displaymath}
\itexarray{
    U &\overset{ \eta^\sharp_U }{\longrightarrow}& \sharp U
    \\
    {}^{\mathllap{i}}\Big\downarrow && \Big\downarrow{}^{\mathrlap{\sharp i}}
    \\
    X &\underset{\eta^\sharp_X}{\longrightarrow}& \sharp X
  }
\end{displaymath}
is a [[pullback]] square.

\end{prop}
\begin{proof}
By the [[universal property]] of a [[pullback]]/[[fiber product]] and the nature of $\sharp$, we have $U \simeq X \times_{\sharp X} \sharp U$ precisely if continuous functions out of some topological space $Z$ into $U$ are in [[natural bijection]] with continuous functions $Z \to X$ whose underlying function $Z \to X \to \sharp X$ factors through the underlying function of $i$. This implies the statement by Prop. \ref{UniversalPropertyOfSupspaceInclusion}.

\end{proof}
\begin{remark}
\label{FormulationInCohesiveHoTT}\hypertarget{FormulationInCohesiveHoTT}{}
\textbf{(formulation in [[cohesive homotopy type theory]])}

The pullback square of the $\sharp$-unit in Prop. \ref{UniversalPropertyViaSharpModality} should correspond (after generalizing from [[topological spaces]] to suitable [[topological ∞-groupoids]]) to the [[categorical semantics]] of what in [[cohesive homotopy type theory]] is the statement that the [[characteristic function]]

\begin{displaymath}
\chi_U \;\colon\; X \to Prop
\end{displaymath}
to the [[universe of propositions]] factors through the universe of [[sharp modality|sharp]]-[[modal types]]. In this form topological subspace inclusions are characterized in \hyperlink{Shulman15}{Shulman 15, Remark 3.14}.

\end{remark}
\hypertarget{general}{}\subsubsection*{{General}}\label{general}

A subspace $i: Y \hookrightarrow X$ is \textbf{closed} if $Y$ is [[closed subset|closed]] as a subset of $X$ (or if $i$ is a [[closed map]]), and is \textbf{open} if $Y$ is open as a subset of $X$ (or if $i$ is an [[open map]]).

\begin{prop}
\label{TopologicalEmbeddingsAreTheRegularMonos}\hypertarget{TopologicalEmbeddingsAreTheRegularMonos}{}
Topological subspace inclusions ([[topological embedding]]) are precisely the [[regular monomorphisms]] in the [[category]] [[Top]] of all [[topological spaces]].

\end{prop}
For example, the [[equalizer]] of two maps $f, g \colon X \stackrel{\to}{\to} Y$ in [[Top]] is computed as the equalizer at the underlying-set level, equipped with the subspace topology.

\begin{lemma}
\label{pushout}\hypertarget{pushout}{}
The [[pushout]] in [[Top]] of any (closed/open) subspace $i \colon A \hookrightarrow B$ along any [[continuous function]] $f \colon A \to C$,

\begin{displaymath}
\itexarray{
A & \stackrel{i}{\hookrightarrow} & B \\ 
\mathllap{f} \downarrow & po & \downarrow \mathrlap{g} \\ 
C & \underset{j}{\hookrightarrow} & D,
}
\end{displaymath}
is a (closed/open) subspace $j: C \hookrightarrow D$.

\end{lemma}
\begin{proof}
Since $U = \hom(1, -): Top \to Set$ is [[faithful functor|faithful]], we have that monos are reflected by $U$; also monos and pushouts are preserved by $U$ since $U$ has both a [[left adjoint]] and a [[right adjoint]]. In $Set$, the pushout of a mono along any map is a mono, so we conclude $j$ is monic in $Top$. Furthermore, such a pushout diagram in $Set$ is also a pullback, so that we have the [[Beck-Chevalley condition|Beck-Chevalley equality]] $\exists_i \circ f^\ast = g^\ast \exists_j \colon P(C) \to P(B)$ (where $\exists_i \colon P(A) \to P(B)$ is the [[direct image]] map between [[power sets]], and $f^\ast: P(C) \to P(A)$ is the [[inverse image]] map).

To prove that $j$ is a subspace, let $U \subseteq C$ be any open set. Then there exists open $V \subseteq B$ such that $i^\ast(V) = f^\ast(U)$ because $i$ is a subspace inclusion. If $\chi_U \colon C \to \mathbf{2}$ and $\chi_V \colon B \to \mathbf{2}$ are the maps to [[Sierpinski space]] that classify these open sets, then by the universal property of the pushout, there exists a unique continuous map $\chi_W \colon D \to \mathbf{2}$ which extends the pair of maps $\chi_U, \chi_V$. It follows that $j^{-1}(W) = U$, so that $j$ is a subspace inclusion.

If moreover $i$ is an open inclusion, then for any open $U \subseteq C$ we have that $j^\ast(\exists_j(U)) = U$ (since $j$ is monic) and (by Beck-Chevalley) $g^\ast(\exists_j(U)) = \exists_i(f^\ast(U))$ is open in $B$. By the definition of the topology on $D$, it follows that $\exists_j(U)$ is open, so that $j$ is an open inclusion. The same proof, replacing the word ``open'' with the word ``closed'' throughout, shows that the pushout of a closed inclusion $i$ is a closed inclusion $j$.

\end{proof}
A similar (but even simpler) line of argument establishes the following result.

\begin{lemma}
\label{transfinite}\hypertarget{transfinite}{}
Let $\kappa$ be an [[ordinal]], viewed as a [[preorder]] category, and let $F: \kappa \to Top$ be a functor that preserves [[directed colimits]]. Then if $F(i \leq j)$ is a (closed/open) subspace inclusion for each morphism $i \leq j$ of $\kappa$, then the canonical map $F(0) \to colim_{i \in \kappa} F(i)$ is also a (closed/open) inclusion.

\end{lemma}
\hypertarget{variations}{}\subsection*{{Variations}}\label{variations}

There is also a notion of a [[Grothendieck topology]] induced along a [[functor]] from a Grothendieck topology on another category (actually the input can be a somewhat more general [[coverage]], then the topology induced along the [[identity functor]] will serve as a sort of a completion). (this will be explained later).

A topology may be induced by more than a [[function]] other than a subset inclusion, or indeed by a family of functions out of $Y$ (not necessarily all with the same [[target]]). However, the term `induced topology' is often (usually?) restricted to subspaces; the general concept is called a [[weak topology]]. (This construction can be done in any [[topological concrete category]]; in this generality it is often called an [[initial structure]] for a [[sink|source]].) The dual construction (involving functions to $Y$) is a [[strong topology]] (or [[final structure]] for a [[sink]]); an example is the [[quotient topology]] on a [[quotient space]].

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

\begin{itemize}%
\item [[embedding]]


\item [[embedding of topological spaces]]


\item [[embedding of smooth manifolds]]



\end{itemize}
[[!include universal constructions of topological spaces -- table]]

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

\begin{itemize}%
\item [[Nicolas Bourbaki]], \emph{Elements of Mathematics -- General topology}, 1971, 1990


\item [[James Munkres]], \emph{Topology}, Prentice Hall (1975, 2000)


\item [[John Terilla]], \emph{Notes on categories, the subspace topology and the product topology} 2014 (\href{https://math.mit.edu/~jhirsh/terilla_subspace_quotient.pdf}{pdf})


\item [[Mike Shulman]], \emph{Brouwer's fixed-point theorem in real-cohesive homotopy type theory}, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (\href{https://arxiv.org/abs/1509.07584}{arXiv:1509.07584}, \href{https://doi.org/10.1017/S0960129517000147}{doi:10.1017/S0960129517000147})



\end{itemize}
[[!redirects subspace topology]] [[!redirects subspace topologies]]

[[!redirects topological subspace]] [[!redirects topological subspaces]]

[[!redirects subspace of a topological space]] [[!redirects subspaces of a topological space]]



\end{document}