Contents

# Contents

## Definition

###### Definition

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

$\left( U_Y \subset Y\,\,\text{open} \right) \,\Leftrightarrow\, \left( \underset{U_X \in \tau_X}{\exists} \left( U_Y = U_X \cap Y \right) \right) \,.$

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

graphics grabbed from Munkres 75

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. Such a map is referred to as a subspace inclusion.

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

## Properties

### Universal property

###### Proposition

(universal property of subspace topology)

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

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

$\array{ Z &\overset{f}{\longrightarrow}& U \\ &{}_{\mathllap{i \circ f}}\searrow& \Big\downarrow{}^{\mathrlap{i}} \\ && X }$

The elementary proof is spelled out, for instance, in Terilla 14, theorem 1. Of course this is just another way to speak of the initial topology.

The universal characterization of Prop. lends itself to formalization via axioms for cohesion:

###### Definition

(sharp modality on topological spaces)

Let

$Set \underoverset {\underset{coDisc}{\longrightarrow}} {\overset{\Gamma}{\longleftarrow}} {\bot} Top$

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

$\sharp \;\coloneqq\; coDisc \circ \Gamma \;\colon\; Top \longrightarrow Top$

for the induced modal operator on Top (sharp modality). We write

$id \overset{\eta^{\sharp}}{\longrightarrow} \sharp$

for the unit morphism of this adjunction.

Notice that this means that for any topological space $Z$, every function of underlying sets

$Z \longrightarrow \sharp X$

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:

$Hom_{Top}( Z, \sharp X ) \;\simeq\; Hom_{Set}(\Gamma(Z), \Gamma(X)) \,.$
###### Proposition

Let $U \overset{i}{\longrightarrow} X$ be an injective continuous function between topological spaces. Then this is a subspace inclusion (Def. ) precisely if its naturality square of the $\sharp$-unit (Def. )

$\array{ U &\overset{ \eta^\sharp_U }{\longrightarrow}& \sharp U \\ {}^{\mathllap{i}}\Big\downarrow && \Big\downarrow{}^{\mathrlap{\sharp i}} \\ X &\underset{\eta^\sharp_X}{\longrightarrow}& \sharp X }$

is a pullback square.

###### 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. .

###### Remark

(formulation in cohesive homotopy type theory)

The pullback square of the $\sharp$-unit in Prop. 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

$\chi_U \;\colon\; X \to Prop$

to the universe of propositions factors through the universe of sharp-modal types. In this form topological subspace inclusions are characterized in Shulman 15, Remark 3.14.

### General

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 $\,$