# Contents

## Definition

Given a topological space $X$ in the sense of (Bourbaki 71) (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 $\tau_Y = \tau_X \cap_{pw} \{Y\} \coloneqq \{ U \cap Y | U\in\tau_X\}$. In other words, $\tau_Y$ is the smallest topology on $Y$ such that the inclusion $Y \hookrightarrow X$ is continuous (the initial topology on that map). 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 continuous function that factors as a homeomorphism onto its image equipped with the subspace topology is called an embedding of topological spaces.

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 $X$.

## Examples

• 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 metric topology is generated from the sub-base $\beta = \{ [0, a) ,(a,1]\}_{a \in [0,1]}$.

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

## Properties

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).

###### Proposition

Topological subspace inclusions (topological embedding) are precisely the regular monomorphisms in the category Top of all topological spaces.

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.

###### Lemma

The pushout in Top of any (closed/open) subspace $i \colon A \hookrightarrow B$ along any continuous function $f \colon A \to C$,

$\array{ A & \stackrel{i}{\hookrightarrow} & B \\ \mathllap{f} \downarrow & po & \downarrow \mathrlap{g} \\ C & \underset{j}{\hookrightarrow} & D, }$

is a (closed/open) subspace $j: C \hookrightarrow D$.

###### Proof

Since $U = \hom(1, -): Top \to Set$ 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 $Set$, 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 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 $j$.

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 $F: \kappa \to Top$ 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.

## 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.

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits$\phantom{AAAA}$colimits
$\,$ point space$\,$$\,$ empty space $\,$
$\,$ product topological space $\,$$\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$$\,$ quotient topological space $\,$
$\,$ fiber space $\,$$\,$ space attachment $\,$
$\,$ mapping cocylinder, mapping cocone $\,$$\,$ mapping cylinder, mapping cone, mapping telescope $\,$
$\,$ cell complex, CW-complex $\,$

## References

Revised on June 21, 2017 04:22:33 by Urs Schreiber (131.220.184.222)