nLab Top

Context

Topology

topology

algebraic topology

Contents

Definition

Top denotes the category whose objects are topological spaces and whose morphisms are continuous functions between them. Its isomorphisms are the homeomorphisms.

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 4 below):

Definition

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. 1, 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. 1, 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. 1 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

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

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 1.

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 2.

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 8 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 9, 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)

Relation with $Set$

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. 1 means in particular that:

Proposition

The category Top has all small limits and colimits. The forgetful functor $U \colon Top \to Set$ from def. 2 preserves and lifts limits and colimits.

(But it does not create or reflect them.)

Proposition

The forgetful functor $U$ from def. 2 has a left adjoint $disc$, given by sending a set $S$ to the corresponding discrete topological space, example 5

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

The forgetful functor $U$ from def. 2 exhibits $Top$ as

Mono-/Epimorphisms, Quotients and Intersections

Proposition

The topological subspace inclusions, example 1, are precisely the regular monomorphisms in $Top$.

Lemma

The pushout in Top of any (closed/open) topological subspace inclusion $i \colon A \hookrightarrow B$, example 1, 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.

References

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