Top (changes) in nLab

Context

Topology

• Definition
• Properties
  • Universal constructions
  • Relation with SetSet
  • Mono-/Epimorphisms
  • Intersections and quotients
  • Closed monoidal structure
• Related concepts
• References

Definition

(weak topology and strong topology)

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

• For {S→f iS i} i∈I\{S \stackrel{f_i}{\to} S_i \}_{i \in I} a set of functions out of SS, the initial topology τ initial({f i} i∈I)\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} i∈I))→X if_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i are continuous.

• For {S i→f iS} i∈I\{S_i \stackrel{f_i}{\to} S\}_{i \in I} a set of functions into SS, the final topology τ final({f i} i∈I)\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} i∈I))f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I})) are continuous.

Example

For XX a single topological space, and ι S:S↪U(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.

Example

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.

Proposition

Let II be a small category and let X •:I⟶TopX_\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 i∈SetS_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⟵ i∈IS 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 } \,.

   Hence

   lim⟵ i∈IX i≃(lim⟵ i∈IS i,τ initial({p i} i∈I)) \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⟶ i∈IS i. \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.

   Hence

   lim⟶ i∈IX i≃(lim⟶ i∈IS i,τ final({ι i} i∈I)) \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)

Proof

The required universal property of (lim⟵ i∈IS i,τ initial({p i} i∈I))\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 S⟶lim⟵ i∈IS 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.

Example

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

Example

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

Example

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

eq(f,g)↪S X⟶g⟶fS 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.

Example

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

S X⟶g⟶fS Y⟶coeq(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 x∈Xx \in X) and equipped with the quotient topology, example 2.

Example

For

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≃(X⊔Y)/∼ 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)

Example

As an important special case of example 9, let

i n:S n−1⟶D 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 ℝ n\mathbb{R}^n).

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

{D n⟵i nS n−1⟶i 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 n−1 ⟶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)

Proof

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 S⊂XS \subset X is an open subset precisely if the pre-images (restrictions)

i A −1(S)=S∩AAAAandAAAi B −1(S)=S∩B 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 S⊂XS \subset X is open, then the intersections A∩S⊂AA \cap S \subset A and B∩S⊂BB \cap S \subset B are open in these subspaces.

Conversely, assume that A∩S⊂AA \cap S \subset A and B∩S⊂BB \cap S \subset B are open. We need to show that then S⊂XS \subset X is open.

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

S =S∩X =S∩(A∪B) =(S∩A)∪(S∩B) \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⊂XA,B \subset X are both closed subsets.

Again by the nature of the subspace topology, that A∩S⊂AA \cap S \subset A and B∩S⊂BB \cap S \subset B are open means that there exist open subsets S A,S B⊂XS_A, S_B \subset X such that A∩S=A∩S AA \cap S = A \cap S_A and B∩S=B∩S BB \cap S = B \cap S_B. Since A,B⊂XA,B \subset X are closed by assumption, this means that A∖S,B∖S⊂XA \setminus S, B \setminus S \subset X are still closed, hence that X∖(A∖S),X∖(B∖S)⊂XX \setminus (A \setminus S), X \setminus (B \setminus S) \subset X are open.

Now observe that (by de Morgan duality)

S =X∖(X∖S) =X∖((A∪B)∖S) =X∖((A∖S)∪(B∖S)) =(X∖(A∖S))∩(X∖(B∖S)). \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.

Definition

Write

U:Top⟶Set 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)=S∈SetU(X) = S \in Set and which regards a continuous function as a plain function on the underlying sets.

Proof

Regarding the first statement: An injective continuous function f:X→Yf \colon X \to Y clearly has the cancellation property that defines monomorphisms: for parallel continuous functions g 1,g 2:Z→Xg_1,g_2 \colon Z \to X, if f∘g 1=f∘g 2f \circ g_1 = f \circ g_2, 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:X→Yi \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.