topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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 TopTop 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).


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 TopTop as a topological concrete category (see Proposition 4 below).


examples of universal constructions of topological spaces:

\, 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 \,



Let {X i=(S i,τ i)Top} iI\{X_i = (S_i,\tau_i) \in Top\}_{i \in I} be a class of topological spaces, and let SSetS \in Set be a bare set. Then

  • For {Sf iS i} iI\{S \stackrel{f_i}{\to} S_i \}_{i \in I} a set of functions out of SS, the initial topology τ initial({f i} iI)\tau_{initial}(\{f_i\}_{i \in I}) is the topology on SS with the minimum collection of open subsets such that all f i:(S,τ initial({f i} iI))X if_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i are continuous.

  • For {S if iS} iI\{S_i \stackrel{f_i}{\to} S\}_{i \in I} a set of functions into SS, the final topology τ final({f i} iI)\tau_{final}(\{f_i\}_{i \in I}) is the topology on SS with the maximum collection of open subsets such that all f i:X i(S,τ final({f i} iI))f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I})) are continuous.


For XX a single topological space, and ι S:SU(X)\iota_S \colon S \hookrightarrow U(X) a subset of its underlying set, then the initial topology τ intial(ι S)\tau_{intial}(\iota_S), def. 1, is the subspace topology, making

ι S:(S,τ initial(ι S))X \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X

a topological subspace inclusion.


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


Let II be a small category and let X :ITopX_\bullet \colon I \longrightarrow Top be an II-diagram in Top (a functor from II to TopTop), with components denoted X i=(S i,τ i)X_i = (S_i, \tau_i), where S iSetS_i \in Set and τ i\tau_i a topology on S iS_i. Then:

  1. The limit of X 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 ip_i which are the limiting cone components:

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


    lim iIX i(lim iIS i,τ initial({p i} iI)) \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 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 ι i\iota_i of the colimiting cocone

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


    lim iIX i(lim iIS i,τ final({ι i} iI)) \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)


The required universal property of (lim iIS i,τ initial({p i} iI))\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) is immediate: for

(S,τ) f i f j X i X j \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 Slim iIS iS \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.


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


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

In particular:


For SSetS \in Set, the SS-indexed coproduct of the point, sS*\underset{s \in S}{\coprod}\ast , is the set SS itself equipped with the final topology, hence is the discrete topological space on SS.


For {X i} iI\{X_i\}_{i \in I} a set of topological spaces, their product iIX iTop\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 SS is a finite set, such as for binary product spaces X×YX \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.


The equalizer of two continuous functions f,g:XYf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the equalizer of the underlying functions of sets

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

(hence the largets subset of S XS_X on which both functions coincide) and equipped with the subspace topology, example 1.


The coequalizer of two continuous functions f,g:XYf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the coequalizer of the underlying functions of sets

S XgfS Ycoeq(f,g) 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)g(x)f(x) \sim g(x) for all xXx \in X) and equipped with the quotient topology, example 2.



A g Y f X \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

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

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

By example 8 the pushout/attaching space is the quotient topological space

X AY(XY)/ X \sqcup_A Y \simeq (X\sqcup Y)/\sim

of the disjoint union of XX and YY subject to the equivalence relation which identifies a point in XX with a point in YY if they have the same pre-image in AA.

(graphics from Aguilar-Gitler-Prieto 02)


As an important special case of example 9, let

i n:S n1D n 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 n\mathbb{R}^n).

Then the colimit in Top under the diagram, i.e. the pushout of i ni_n along itself,

{D ni nS n1i nD n}, \left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,,

is the n-sphere S nS^n:

S n1 i n D n i n (po) D n 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)


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

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

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

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

Write i A:AXi_A \colon A \to X and i B:BXi_B \colon B \to X for the corresponding inclusion continuous functions.

Then the commuting square

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

is a pushout square in TopTop (example 9).

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

f:XY f \;\colon\; X \longrightarrow Y

is a continuous function as soon as its two restrictions

f| A:AYAAAAf| A:BY f\vert_A \;\colon\; A \longrightarrow Y \phantom{AAAA} f\vert_A \;\colon\; B \longrightarrow Y

are continuous.


Clearly the underlying diagram of underlying sets is a pushout in Set. Therefore by prop. 1 we need to show that the topology on XX is the final topology induced by the set of functions {i A,i B}\{i_A, i_B\}, hence that a subset SXS \subset X is an open subset precisely if the pre-images (restrictions)

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

are open subsets of AA and BB, respectively.

Now by definition of the subspace topology, if SXS \subset X is open, then the intersections ASAA \cap S \subset A and BSBB \cap S \subset B are open in these subspaces.

Conversely, assume that ASAA \cap S \subset A and BSBB \cap S \subset B are open. We need to show that then SXS \subset X is open.

Consider now first the case that A;BXA;B \subset X are both open open. Then by the nature of the subspace topology, that ASA \cap S is open in AA means that there is an open subset S AXS_A \subset X such that AS=AS AA \cap S = A \cap S_A. Since the intersection of two open subsets is open, this implies that AS AA \cap S_A and hence ASA \cap S is open. Similarly BSB \cap S. Therefore

S =SX =S(AB) =(SA)(SB) \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,BXA,B \subset X are both closed subsets.

Again by the nature of the subspace topology, that ASAA \cap S \subset A and BSBB \cap S \subset B are open means that there exist open subsets S A,S BXS_A, S_B \subset X such that AS=AS AA \cap S = A \cap S_A and BS=BS BB \cap S = B \cap S_B. Since A,BXA,B \subset X are closed by assumption, this means that AS,BSXA \setminus S, B \setminus S \subset X are still closed, hence that X(AS),X(BS)XX \setminus (A \setminus S), X \setminus (B \setminus S) \subset X are open.

Now observe that (by de Morgan duality)

S =X(XS) =X((AB)S) =X((AS)(BS)) =(X(AS))(X(BS)). \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 SS as the intersection of two open subsets, hence as open.


If X,Y,ZX, Y, Z are normal topological spaces and h:XZh: X \to Z is a closed embedding of topological spaces and f:XYf: X \to Y is a continuous function, then in the pushout diagram in TopTop (example 9)

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

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

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

Relation with SetSet

Write Set for the category of sets.



U:TopSet U \colon Top \longrightarrow Set

for the forgetful functor that sends a topological space X=(S,τ)X = (S,\tau) to its underlying set U(X)=SSetU(X) = S \in Set and which regards a continuous function as a plain function on the underlying sets.

Prop. 1 means in particular that:


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

(But it does not create or reflect them.)


The forgetful functor UU from def. 2 has a left adjoint discdisc, given by sending a set SS to the corresponding discrete topological space, example 5

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

The forgetful functor UU from def. 2 exhibits TopTop as



(regular monomorphisms of topological spaces)

In the category Top of topological space,

  1. the monomorphisms are the 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).


Regarding the first statement: An injective continuous function f:XYf \colon X \to Y clearly has the cancellation property that defines monomorphisms: for parallel continuous functions g 1,g 2:ZXg_1,g_2 \colon Z \to X: if fg 1=fg 1f \circ g_1 = f \circ g_1, then g 1=g 2g_1 = g_2 because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if ff has the cancellation property, then testing on points g 1,g 2:*Xg_1, g_2 \colon \ast \to X gives that ff is injective.

Regarding the second statement: from the construction of equalizers in Top (example 7) we have that these are topological subspace inclusions.

Conversely, let i:XYi \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)(i_1, i_2) by taking the pushout of ii 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 ii back, and again by example 7, it gets equipped with the subspace topology. This completes the proof.

Intersections and quotients


The pushout in Top of any (closed/open) topological subspace inclusion i:ABi \colon A \hookrightarrow B, example 1, along any continuous function f:ACf \colon A \to C is itself an a (closed/open) subspace j:CDj \colon C \hookrightarrow D.

For proof see there.


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

See also

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

An axiomatic desciption of TopTop 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)

category: category

Revised on June 14, 2017 04:02:29 by Urs Schreiber (