nLab
subspace topology

Definition

Given a topological space $X$ in the sense of Bourbaki (that is, a set $X$ and a topology ${\tau }_X$) and a subset $Y$ of $X$, a topology ${\tau }_Y$ on $Y$ is said to be the topology induced from ${\tau }_X$ by the set inclusion $Y\hookrightarrow X$ if ${\tau }_Y={\tau }_X{\cap }_{\mathrm{pw}}\{Y\}≔\{U\cap Y\mid U\in {\tau }_X\}$. In other words, ${\tau }_Y$ is the smallest topology on $Y$ such that the inclusion $Y\hookrightarrow X$ is continuous. The pair $(Y,{\tau }_Y)$ is then said to be a topological subspace of $(X,{\tau }_X)$. The induced topology is for that reason sometimes called the subspace topology on $Y$.

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

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

Properties

Proposition

Subspace inclusions are precisely the regular monomorphisms in $\mathrm{Top}$.

For example, the equalizer of two maps $f,g:X\overrightarrow{\to }Y$ in $\mathrm{Top}$ is computed as the equalizer at the underlying-set level, equipped with the subspace topology.

Lemma

The pushout in $\mathrm{Top}$ of any (closed/open) subspace $i:A\hookrightarrow B$ along any continuous map $f:A\to C$ is a (closed/open) subspace $j:C\hookrightarrow D$.

Proof

Since $U=\mathrm{hom}(1,-):\mathrm{Top}\to \mathrm{Set}$ is 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 $\mathrm{Set}$, the pushout of a mono along any map is a mono, so we conclude $j$ is monic in $\mathrm{Top}$. Furthermore, such a pushout diagram in $\mathrm{Set}$ is also a pullback, so that we have the Beck-Chevalley equality ${\exists }_i\circ f^{*}=g^{*}{\exists }_j:P(C)\to P(B)$ (where ${\exists }_i:P(A)\to P(B)$ is the direct image map between power sets, and $f^{*}:P(C)\to P(A)$ 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 $i^{*}(V)=f^{*}(U)$ because $i$ is a subspace inclusion. If ${\chi }_U:C\to 2$ and ${\chi }_V:B\to 2$ 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 ${\chi }_W:D\to 2$ 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 $j^{*}({\exists }_j(U))=U$ (since $j$ is monic) and (by Beck-Chevalley) $g^{*}({\exists }_j(U))={\exists }_i(f^{*}(U))$ 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 $j$.

A similar (but even simpler) line of argument establishes the following result.

Lemma

Let $\kappa$ be an ordinal, viewed as a preorder category, and let $F:\kappa \to \mathrm{Top}$ be a functor that preserves directed colimits. Then if $F(i\le j)$ is a (closed/open) subspace inclusion for each morphism $i\le j$ of $\kappa$, then the canonical map $F(0)\to {\mathrm{colim}}_{i\in \kappa }F(i)$ is also a (closed/open) inclusion.

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 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.