Contents

# Contents

## Definition

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 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. 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 classical model structure on topological spaces. The simplicial localization of Top at the weak homotopy equivalences is the archetypical (∞,1)-category, equivalent to ∞Grpd (see at homotopy hypothesis).

## Properties

### Universal constructions

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 below).

$\,$

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 $\,$

$\,$

###### Definition

(weak topology and strong topology)

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

• For $\{S \stackrel{f_i}{\to} S_i \}_{i \in I}$ a set of functions out of $S$, the 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.

• For $\{S_i \stackrel{f_i}{\to} S\}_{i \in I}$ a set of functions into $S$, the 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.

###### Example

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. , is the subspace topology, making

$\iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X$

a topological subspace inclusion.

###### Example

Conversely, for $p_S \colon U(X) \longrightarrow S$ an epimorphism, then the final topology $\tau_{final}(p_S)$ on $S$ is the quotient topology.

###### Proposition

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:

1. The limit of $X_\bullet$ exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology, def. , for the functions $p_i$ which are the limiting cone components:

$\array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,.$

Hence

$\underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$
2. 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. for the component maps $\iota_i$ of the colimiting cocone

$\array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.$

Hence

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

(e.g. Bourbaki 71, section I.4)

###### 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

$\array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_j }$

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. But this is precisely what the initial topology ensures.

The case of the colimit is formally dual.

###### Example

The limit over the empty diagram in $Top$ is the point $\ast$ with its unique topology.

###### Example

For $\{X_i\}_{i \in I}$ a set of topological spaces, their coproduct $\underset{i \in I}{\sqcup} X_i \in Top$ is their disjoint union.

In particular:

###### Example

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 $S$.

###### Example

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 product topology, also called the Tychonoff product.

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

###### Example

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

$eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y$

(hence the largets subset of $S_X$ on which both functions coincide) and equipped with the subspace topology, example .

###### Example

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

$S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g)$

(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 .

###### Example

For

$\array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X }$

two continuous functions out of the same domain, then the colimit under this diagram is also called the pushout, denoted

$\array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,.$

(Here $g_\ast f$ is also called the pushout of $f$, or the cobase change of $f$ along $g$.) If $g$ is an inclusion, one also write $X \cup_f Y$ and calls this the attaching space.

By example the pushout/attaching space is the quotient topological space

$X \sqcup_A Y \simeq (X\sqcup Y)/\sim$

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 Aguilar-Gitler-Prieto 02)

###### Example

As an important special case of example , let

$i_n \colon S^{n-1}\longrightarrow D^n$

be the canonical inclusion of the standard (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,

$\left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,,$

is the n-sphere $S^n$:

$\array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,.$

(graphics from Ueno-Shiga-Morita 95)

###### Example

(union of two open or two closed subspaces is pushout)

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

1. $A,B \subset X$ are both open subsets or are both closed subsets;

2. they constitute a cover: $X = A \cup B$

Write $i_A \colon A \to X$ and $i_B \colon B \to X$ for the corresponding inclusion continuous functions.

Then the commuting square

$\array{ A \cap B &\longrightarrow& A \\ \downarrow && \downarrow^{\mathrlap{i_A}} \\ B &\underset{i_B}{\longrightarrow}& X }$

is a pushout square in $Top$ (example ).

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

$f \;\colon\; X \longrightarrow Y$

is a continuous function as soon as its two restrictions

$f\vert_A \;\colon\; A \longrightarrow Y \phantom{AAAA} f\vert_A \;\colon\; B \longrightarrow Y$

are continuous.

###### Proof

Clearly the underlying diagram of underlying sets is a pushout in Set. Therefore by prop. we need to show that the 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)

$i_A^{-1}(S) = S \cap A \phantom{AAA} \text{and} \phantom{AAA} i_B^{-1}(S) = S \cap B$

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{aligned} S & = S \cap X \\ & = S \cap (A \cup B) \\ & = (S \cap A) \cup (S \cap B) \end{aligned}

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{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}

This exhibits $S$ as the intersection of two open subsets, hence as open.

###### Example

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 )

$\array{ X & \stackrel{h}{\to} & Z \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\ Y & \underset{k}{\to} & W, }$

the space $W$ is normal and $k: Y \to W$ is a closed embedding.

For proof of this and related statements see at colimits of normal spaces.

### Relation with $Set$

Write Set for the category of sets.

###### Definition

Write

$U \colon Top \longrightarrow Set$

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.

Prop. means in particular that:

###### Proposition

The category Top has all small limits and colimits. The forgetful functor $U \colon Top \to Set$ from def. preserves and lifts limits and colimits.

(But it does not create or reflect them.)

###### Proposition

The forgetful functor $U$ from def. has a left adjoint $disc$, given by sending a set $S$ to the corresponding discrete topological space, example

$Top \stackrel{\overset{disc}{\longleftarrow}}{\underset{U}{\longrightarrow}} Set \,.$
###### Proposition

The forgetful functor $U$ from def. exhibits $Top$ as

### Mono-/Epimorphisms

###### Proposition

(regular monomorphisms of topological spaces)

In the category Top of topological spaces,

1. the monomorphisms are those continuous functions which are injective functions;

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

###### 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 $f \circ g_1 = f \circ g_2$, 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 ) 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 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 , it gets equipped with the subspace topology. This completes the proof.

### Intersections and quotients

###### Lemma

The pushout in Top of any (closed/open) topological subspace inclusion $i \colon A \hookrightarrow B$, example , along any continuous function $f \colon A \to C$ is itself an a (closed/open) subspace $j \colon C \hookrightarrow D$.

For proof see there.

### Closed monoidal structure

It is well known that Top is not cartesian closed (see for example at convenient category of topological spaces).

It is however closed monoidal.

• The tensor product $X\otimes Y$ is given by the cartesian product of the underlying spaces, equipped with the topology of separate continuity, formed by the sets $U\subseteq X\times Y$ such that

$U_x \;\coloneqq\; \{y\in Y : (x,y\in U)\}$

is an open subset of $Y$ and

$U_x \;\coloneqq\; \{x\in X : (x,y)\in U\}$

is an open subset of $X$. Equivalently, it is the topology such that for all spaces $Z$, a function $f:X\otimes Y\to Z$ is continuous if and only if: for every $x\in X$ the function $y\mapsto f(x,y)$ is continuous, and for every $y\in Y$, the functions $x\mapsto f(x,y)$ is continuous.

• The internal hom $[X,Y]$ is given by the set of continuous functions $X\to Y$, together with the topology of pointwise convergence, generated by the (sub-basic) sets

$S(x,V) \;\coloneqq\; \{f:X\to Y : f(x)\in V\}$

for each $x\in X$ and each open $V\subseteq Y$. Equivalently, a net $(f_\alpha:X\to Y)$ tends to $f:X\to Y$ if and only if for all $x\in X$, $f_\alpha(x)\to f(x)$ in $Y$.

## References

For general references see those listed at topology, such as

• Nicolas Bourbaki, chapter 1 Topological Structures of Elements of Mathematics III: General topology, Springer 1971, 1990

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 12 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

An axiomatic desciption of $Top$ along the lines of ETCS for Set is discussed in

• Dana Schlomiuk, An elementary theory of the category of topological space, Transactions of the AMS, volume 149 (1970)

For its closed monoidal structure, see:

• Maria Cristina Pedicchio and Fabio Rossi, Monoidal closed structures for topological spaces: counter-example to a question of Booth and Tillotson, Cahiers de topologie et géométrie différentielle catégoriques, 24(4), 1983.
• Appendix A of Paolo Perrone and Ruben Van Belle, Convergence of martingales via enriched dagger categories, 2024. (arXiv)
category: category

Last revised on July 26, 2024 at 09:48:17. See the history of this page for a list of all contributions to it.