13. A little point-set topology.

∗ ∗ ∗ Open and closed sets ∗ ∗ ∗

13.1 Definition. A subset τ of the power set P(X) of a set X is called a topology on X if the following axioms hold: (O1) ∅ ∈ τ and X ∈ τ (O2) S, T ∈ τ ⟹ S ∩ T ∈ τ (O3) S_{i} ∈ τ for all i ∈ I (I an arbitrary index set) ⟹ ⋃_{i∈I} S_{i} ∈ τ (X, τ) is then called a topological space. The members of τ are called open sets.

If (X, τ) is a topological space, then the complements of open sets, A = X \ S with S ∈ τ, are called closed sets. The collection of closed sets C = {X \O | O ∈ τ} satisfies (C1) ∅ ∈ C and X ∈ C. (C2) A, B ∈ C ⟹ A ∪ B ∈ C. (C3) A_{i} ∈ C for all i ∈ I (I an arbitrary index set) ⟹ ⋂_{i∈I} A_{i} ∈ C

Conversely, if C ⊆ P(X) satisfys C1–C3, then the complements of elements of C form a topology on X, whose closed sets are precisely the elements of C.

13.2 Example: Zariski topology on the spectrum of a ring. Let R be a commutative ring. A topology on the spectrum of R (Spec(R) = {P | P a prime ideal of R}) is defined by specifying its closed sets as sets of prime ideals of the form

V (I) = {P ∈ Spec(R) | P ⊇ I},

for some ideal I of R.

There’s nothing to prevent sets from being open and closed at the same time. Sets both open and closed are often called clopen.

13.3 Definition. Let (X, τ) be a topological space.

A collection B of open sets such that every open set is a union of elements of B is called a basis of the topology τ.

A collection S of open sets such that every open set is a union of finite intersections of elements of S is called a subbasis of the topology τ.

Topologies can be defined by specifying a basis or subbasis: If B ⊆ P(X) is closed with respect to finite intersections, then the unions of (arbitrarily many) members of B form a topology on X, of which B is a basis.

If S ⊆ P(X) is any collection of subsets of X, then arbitrary unions of finite intersections of members of S form a topology on X, of which S is a subbasis.

13.4 Example: Order Topology. If (X, ≤) is a totally ordered set, then order topology on X is defined by specifying “open rays”, i.e., sets of the form (a, ∞) = {x ∈ X | a < x} and (−∞, b) = {x ∈ X | x < b}, for a, b ∈ X as a subbasis.

If (X, ≤) has neither a maximal nor a minimal element, then “open intervals” (a, b) = {x ∈ X | a < x < b} form a basis of order topology. (If X does have a maximal or minimal element, then open rays of the form (a, ∞), or (−∞, b), respectively, have to be added to the open intervals to get a basis.)

13.5 Definition. If (X, τ) is a topological space and Y a subset of X then Y inherits a topological structure from X (called subspace topology) through the convention: a subset U of Y is open (in Y ) iff there exists an open subset O of X with U = O ∩ Y .

If Y is an open subset of X then U ⊆ Y is open in Y if and only it is open in X; if Y is a closed subset of X then A ⊆ Y is closed in Y if and only it is closed in X.

13.6 Remark: If (X, ≤) is a totally ordered set and Y ⊆ X, then Y inherits a topology from the order topology of X, and at the same time Y inherits an order relation from X which makes (Y, ≤) a totally ordered set, for which order topology may be defined. These two topologies on Y in general do not agree. (Examples can be found among subsets of the real numbers.)

∗ ∗ ∗ Neighborhoods and neighborhood bases ∗ ∗ ∗

Perhaps a more intuitive approach to topology is through neighborhoods of a point, which (as sets containing an open ball around the point) are already familiar from the study of metric spaces.

13.7 Definition. (∗) Let (X, τ) be a topological space and p ∈ X. A neighborhood of p is a set U such that there exists an open set O with p ∈ O ⊆ U. The set of all neighborhoods of a point p is called the neighborhood filter of p. We will denote it by U(p).

13.8 Remark: Let (X, τ) be a topological space and p ∈ X. The neighborhood filter of p has the properties (U1) ∀U ∈ U(p) p ∈ U. (U2) U, V ∈ U(p) ⟹ U ∩ V ∈ U(p) (U3) U ∈ U(p) and V ⊇ U ⟹ V ∈ U(p) (U4) ∀U ∈ U(p) ∃V ∈ U(p) ∀v ∈ V U ∈ U(v). Also,

(U)  O ∈ τ ⟺ ∀p ∈ O ∃U ∈ U(p) U ⊆ O

Conversely, given a set X and for each p ∈ X a set U(p) ⊆ P(X) such that U1–U4 hold, we can define a topolgy τ on X by (U), and, what is more, the neighborhood filter of each point in the resulting topology τ is exactly the U(p) we started out with.

As with metric spaces, it suffices to know a system of “basic” neighborhoods of a point – with the property that every neighborhood contains one of them – to know all neighborhoods.

13.9 Definition. Let (X, τ) be a topological space. A collection B(p) ⊆ U(p) of neighborhoods of p is called a neighborhood basis of p if for every U ∈ U(p) there exists B ∈ B(p) with B ⊆ U.

If (X, τ) is a topological space, and for each p ∈ X, B(p) is a neighborhood basis, then, for every p ∈ X (B1) ∀B ∈ B(p) p ∈ B. (B2) U, V ∈ B(p) ⟹ ∃B ∈ B(p) B ⊆ U ∩ V (B3) ∀U ∈ B(p) ∃V ∈ B(p) ∀v ∈ V ∃B ∈ B(v) B ⊆ U).

Also,

(B)  O ∈ τ ⟺ ∀p ∈ O ∃B ∈ B(p) B ⊆ O

Conversely, if we are given for every p ∈ X a collection B(p) of subsets of X satisfying B1–B3, we can define a topology on X by (B), and in this topology B(p) will be a neighborhood basis of p.

13.10 Definition. A topological space in which every point has a countable neighborhood basis is said to satisfy the first countability axiom.

13.11 Example: Let (X, d) be a metric space, and B_{ε}(p) = {x ∈ X | d(p, x) < ε} (for ε > 0, p ∈ X) the open ball of radius ε around p. If we define B(p) as the set of all B_{ε}(p) with ε > 0, then B1–B3 hold. In other words, every metric induces a topology in which the open ε-balls with center p form a neighborhood basis of p. Actually, countably many balls B_{\frac{1}{n}} (p), n ∈ ℕ, already form a neighborhood basis of p in this topology. We see that every metric space satisfies the first countability axiom.

13.12 Example: I-adic topology Let R be a commutative ring and I an ideal of R. I-adic topology on R is defined by specifying a neighborhood base of r ∈ R:

B(r) = {r + I^{n} | n ∈ ℕ}.

Note that these basic neighborhoods are both open and closed.

∗ ∗ ∗ Closure and interior ∗ ∗ ∗

13.13 Definition. Let A be a subset of a topological space X. The closure of A, denoted Ā, is defined to be the intersection of all closed sets containing A.

Clearly, Ā is a closed set containing A and every closed set B that contains A also contains Ā.

The closure operator A ↦ Ā on P(X) has the properties: (A1) ¯∅ = ∅. (A2) A ⊆ Ā. (A3) ¯ Ā = Ā. (A4) Ā ∪ B̄ = \overline{A ∪ B}.

Conversely, any operator A ↦ Ā on the power set of a set X with properties A1–A4 may be used to define a topology on X by declaring the closed sets to be precisely the sets of the form Ā for some A ∈ P(X). (To see that these sets satisfy C3, first note that A4 implies A ⊆ C ⟹ Ā ⊆ C̄. From this we get that \overline{⋂_{i∈I} Ā_{i}} ⊆ Ā_{i} for all i ∈ I, i.e., \overline{⋂_{i∈I} Ā_{i}} ⊆ ⋂_{i∈I} Ā_{i} and the reverse inclusion is just A2.)

13.14 Definition. Let (X, τ) be a topological space and A ∈ P(X) then the open interior (or interior for short) of A, denoted A^{∘}, is the union of all open sets contained in A.

Clearly, A^{∘} is open and A^{∘} ⊆ A. Also, (X \ A)^{∘} = X \ Ā. (Note that the interior of a non-empty set may be empty.)

∗ ∗ ∗ Boundry points, accumulation points and such ∗ ∗ ∗

The following properties of the closure of a set (which we defined as the intersection of all closed sets containing it) and the interior of a set (which we defined as the union of all open sets contained in it) are often used as definitions:

13.15 Lemma.

• Ā consists of precisely those points p ∈ X such that U ∩ A ≠ ∅ for every neighborhood U ∈ U(p) and

• A^{∘} consists of precisely those points p∈X such that there exists a neighborhood U ∈ U(p) with U ⊆ A.

13.16 Definition. Let (X, τ) be a topological space, p ∈ X and A ⊆ X then

• p is called a boundary point of A if for every neighborhood U ∈ U(p) both A ∩ U ≠ ∅ and (X \ A) ∩ U ≠ ∅. The set of all boundary points of A is the boundary of A, denoted by δA.

• p is called an interior point of A if p ∈ A^{∘}, i.e., if there exists a neighborhood U ∈ U(p) with U ⊆ A.

• p is called an accumulation point of A if every neighborhood of p contains an element of A other than p.

• p is called an isolated point of A if there exists a neighborhood U of p with U ∩ A = {p}.

Note that interior points of A and isolated points of A are necessarily in A, while boundary points and accumulation points may or may not belong to A.

By purely logical arguments we see that every set A induces a partition of X into three disjoint parts in two different ways ( ˙∪ denotes disjoint union):

(i) X = A^{∘} ˙∪ δA ˙∪ (X \ A)^{∘} and (ii) X = {isolated points of A} ˙∪{accumulation points of A} ˙∪(X \ A)^{∘}.

Also, (iii) Ā = A^{∘} ˙∪ δA and (iv) Ā = {isolated points of A} ˙∪ {accumulation points of A} The last two partitions of Ā follow from the characterization of Ā as the set

of those p such that A ∩ U ≠ ∅ for every U ∈ U(p). They are incomparable in the sense that all four combinations of belonging to one set of one partition and one set of the other are possible, an accumulation point of A can be either an interior point or a boundary point of A, etc. Also, we have seen that

(v) Ā = A ∪ δA and (vi) Ā = A ∪ {accumulation points of A} Unlike (iii) and (iv), the unions (v) and (vi) are in general not disjoint.

∗ ∗ ∗ Continuous functions ∗ ∗ ∗

13.17 Definition. Let (X, τ) and (Y, τ′) be topological spaces. A function f: X →Y is called continuous if f^{−1}(O) is open for every open set O ⊆ Y .

13.18 Remark: Inverse image commutes with arbitrary unions and intersections

f^{−1}(⋃_{i∈I} S_{i}) = ⋃_{i∈I} f^{−1}(S_{i}) and f^{−1}(⋂_{i∈I} S_{i}) = ⋂_{i∈I} f^{−1}(S_{i}).

Therefore, for f: X → Y to be continuous, it suffices that f^{−1}(O) be open for all O in some fixed subbasis of Y . Also,

f^{−1}(Y \ S) = X \ f^{−1}(S).

Therefore, f: X → Y is continuous if and only if f^{−1}(A) is closed for every closed set A ⊆ Y .

In terms of neighborhoods, a function f: X → Y is continuous, if and only if for every x ∈ X, for every U ∈ U(f(x)) there exists a V ∈ U(x) with f(V) ⊆ U. The familiar ε-δ-definition of continuous functions is easily seen to be the specialization to metric spaces of this topological characterization.

13.19 Definition. Let (X, τ) and (Y, τ′) be topological spaces. A function f: X →Y is called open if f(O) is open for every open set O ⊆ X.

A bijective function both open and continuous is called a homeomorphism.

A topology τ_{1} on X is called stronger (or finer) than another topology τ_{2} on the same set X if τ_{1} ⊇ τ_{2} (every τ_{2}-open set is τ_{1}-open); τ_{2} is then called weaker or coarser than τ_{1}. Two trivial topologies exist on every set X: discrete topology τ = P(X) (the finest topology on X) and chaotic topology τ = {∅, X} (the coarsest topology on X).

If τ_{1} and τ_{2} are two topologies on a set X then τ_{1} is stronger than τ_{2} iff id_{X} : (X, τ_{1}) → (X, τ_{2}) is continuous; τ_{1} is weaker than τ_{2} iff id_{X} : (X, τ_{1}) → (X, τ_{2}) is open.

∗ ∗ ∗ Connectedness ∗ ∗ ∗

13.20 Definition. A topological space X is connected, if, whenenver O_{1} and O_{2} are open sets with O_{1} ∪ O_{2} = X and O_{1} ∩ O_{2} = ∅, it follows that O_{1} = ∅ or O_{2} = ∅. A subset Y of X is connected if it is conneced in subspace topology.

13.21 Exercise. If X is connected and f: X → Y continuous, then f(X) is connected.

13.22 Lemma. If X_{i} is a connected subset of X for every i ∈ I (I an arbitrary index set) and ⋂_{i∈I} X_{i} ≠ ∅ then ⋃_{i∈I} X_{i} is connected.

Proof. Easy exercise. □

13.23 Lemma and Definition. The following relation ∼ is an equivalence relation on X: x ∼ y if and only if there exists a connected subset C of X with x, y ∈ C.

The equivalence classes with respect to ∼ are called the connected components of X.

From the above definition it is clear that the connected components of X form a partition of X. Also, the component of x ∈ X is the union of all connected subsets of X containing x, and it is therefore the unique largest connected subset of X containing x.

13.24 Lemma. If Y is a connected subset of X then every set C with Y ⊆ C ⊆ \overline{Y} is connected.

Proof. Exercise. □

13.25 Definition. A topological space X is locally connected if every x ∈ X has a neighborhood basis consisting of connected neighborhoods.

13.26 Lemma. The connected components of a topological space X are closed sets. If every x ∈ X has a connected neighborhood (in particular, if X is locally connected) then they are also open.

Proof. By the lemma above, the closure of a connected component is again connected and therefore contained in the component. If a point x possesses a connected open neighborhood U_{x} then the compoment of x (being the union of all connected sets containing x) contains U_{x}. □

13.27 Definition. A topological space X is totally disconnected if it doesn’t contain any connected set with more than one element; equivalently, if its connected components are singletons.

∗ ∗ ∗ Filters ∗ ∗ ∗

13.28 Definition. Let X be a set. A filter on X is a set F ⊆ P(X) with the properties (1) ∅ ∉ F

(2) If A, B ∈ F then A ∩ B ∈ F.

(3) If A ∈ F and X ⊇ C ⊇ A then C ∈ F.

13.29 Definition. Let F and G be filters on X. We say that F is finer than G (or, equivalently, G is coarser than F) if F ⊇ G.

13.30 Definition. An ultrafilter on X is a filter F with the property: ∀S ⊆ X : S ∈ F ∨ (X \ S) ∈ F.

13.31 Definition. If F is a filter on X then a subset F′ of F is called a filter base of F if ∀F ∈ F ∃F′ ∈ F′ with F′ ⊆ F.

13.32 Remark: If F is a filter then F′ ⊆ F is a filter base of F if and only if F consists precisely of all supersets of elements of F′. We can easily give a criterion for a collection of sets F′ ⊆ P(X) to be the base of a filter on X. The set of supersets of elements of F′ is a filter if and only if F′ has the finite intersection property, that is, the intersection of any finite number of sets in F′ is non-empty.

13.33 Lemma. If F is a filter on X containing neither S nor X \ S, then there exists a filter F_{1} ⊇ F ∪ {S} and a filter F_{2} ⊇ F ∪ {(X \ S)}.

Proof. No F ∈ F is contained in S (otherwise S would be in F), and likewise, no F ∈ F is contained in X \S. Therefore F ∩(X \S) ≠ ∅ for all F ∈ F and F ∩S ≠ ∅ for all F ∈ F. Now take F ∪ {S} as a filter base for F_{1} and F ∪ {(X \ S)} as a filter base for F_{2}. □

13.34 Lemma. A filter is maximal with respect to refinement (i.e., inclusion) if and only if it is an ultrafilter.

Proof. It is clear that no sets can be added to an ultrafilter without violating proprty (1) or (2) in the definition of a filter. Conversely, 13.33 shows that a filter that is not an ultrafilter has a proper refinement. □

13.35 Lemma. For every filter F on X there exists an ultrafilter on X finer than F. F is the intersection of all ultrafilters on X finer than F.

Proof. Consider the set S of all filters on X containing F, ordered by inclusion. Since the union of a chain of filters is again a filter, every chain in S has an upper bound in S. By Zorn’s Lemma, there exists a maximal element in S, which is an ultrafilter containing F, by 13.34. By 13.33, the intersection of all ultrafilters containing F contains no other sets than the elements of F. □

∗ ∗ ∗ Nets ∗ ∗ ∗

13.36 Definition. A directed set is a set I With a binary relation ≤ such that ∀i, j, k ∈ I

(1) i ≤ i.

(2) if i ≤ j and j ≤ k then i ≤ k.

(3) ∃n ∈ I with n ≥ i and n ≥ j.

Note that we do not require anti-symmery.

13.37 Definition. A net in X is a function from a directed set to X, ψ: I → X, usually written (like a sequence) as a list of values indexed by arguments, (x_{i})_{i∈I}.

13.38 Definition. Let ψ: I → X, written as (x_{i})_{i∈I}, be a net in X, J a directed set and and φ: J → I an increasing function that is cofinal in I, that is,

∀i, i′ ∈ I (i ≤ i′ ⟹ φ(i) ≤ φ(i′))

∀i ∈ I ∃k ∈ K : φ(k) ≥ i.

Then the composition of maps ψ ∘ φ: J → X is called a subnet of (x_{i})_{i∈I}, and is written (x_{i_{j}})_{j∈J}.

13.39 Definition. Let (x_{i})_{i∈I} be a net in X and S ⊆ X. We say that (x_{i}) is eventually in S if there exists n ∈ I such that x_{i} ∈ S for all i ≥ n. We say that (x_{i}) is frequently in S, if for all n ∈ I there exists i ∈ I with i ≥ n and x_{i} ∈ S.

13.40 Remark: A set of the form {x_{i} | i ≥ n} for some n ∈ I is called a tail of the net (x_{i})_{i∈I}. A net is eventually in a set S if and only if some tail is contained in S; it is frequently in S if and only if all its tails intersect S nontrivially.

13.41 Definition. An ultranet in X is a net such that for every subset S of X, the net is eventually in S or eventually in X \ S.

13.42 Exercise. Let f: X → Y be any function. If (x_{i})_{i∈I} is an ultranet in X then (f(x_{i}))_{i∈I} is an ultranet in Y . If F is an ultrafilter on X then {f(F) | F ∈ F} is an ultrafilter on f(X) and {S ⊆ Y | ∃F ∈ F : f(F) ⊆ S} is an ultrafilter on Y .

∗ ∗ ∗ Convergence ∗ ∗ ∗

13.43 Definition. Let F be a filter on X, x ∈ X, and U(x) the neighborhood filter of x.

F converges to x if and only if U(x) ⊆ F. In this case, x is called a limit point of F. x is a cluster point of F if and only if F ∩ U ≠ ∅ for all F ∈ F and all U ∈ U(x) (or equivalently, if x ∈ F̄ for all F ∈ F).

13.44 Proposition. Let G ⊆ F be filters on X and x ∈ X.

(1) If G converges to x then the finer filter F converges to x.

(2) If x is a cluster point of F then x is a cluster point of the coarser filter G.

Proof. Follows immediately from the definition of filter convergence and cluster points. □

13.45 Definition. Let x ∈ X and U(x) the neighborhood filter of x.

A net in X converges to x if and only if for every U ∈ U(x), the net is eventually in U. In this case, x is called a limit point of the net. x is a cluster point of a net on X if and only if for every U ∈ U(x), the net is frequently in U.

13.46 Proposition. Let (x_{n_{k}}) be a subnet of the net (x_{n}) on X and x ∈ X. If (x_{n}) converges to x then the subnet (x_{n_{k}}) converges to x. If x is a cluster point of the subnet (x_{n_{k}}) then x is a cluster point of (x_{n}).

Proof. Follows immediately from the definition of subnet. □

13.47 Proposition. An ultrafilter converges against each of its cluster points. Similarly, an ultranet converges against each of its cluster points.

Proof. Suppose F is a filter and x ∈ X such that for all F ∈ F and all U ∈ U(x), U ∩ F ≠ ∅. Then for all U ∈ U(x), (X \ U) ∉ F. If F is an ultrafilter, U ∈ F for all U ∈ U(x) follows. The case of nets is similar. □

The following constructions of a net from a filter and a filter from a net often allow to translate statements about filters to statements about nets and vice versa:

13.48 Lemma and Definition. Let (x_{n})_{n∈N} be a net on X and x ∈ X. The filter constructed from (x_{n})_{n∈N} is defined by taking the set of ends {x_{n} | n ≥ n_{0}} for n_{0} ∈ N as a filter basis.

Then the filter constructed from (x_{n}) converges to x if and only if (x_{n}) converges to x. Also, x is a cluster point of the filter constructed from (x_{n}) if and only if x is a cluster point of (x_{n}).

Proof. Easy exercise. □

13.49 Lemma and Definition. Let F be a filter on X and x ∈ X. The net constructed from F is indexed by the set I = {(F, y) | F ∈ F, y ∈ F } with (F, y) ≥ (F′, y′) :⟺ F ⊆ F′; and x_{(F,y)} = y.

Then the net constructed from F converges to x if and only if F converges to x. Also, x is a cluster point of the net constructed from F if and only if x is a cluster point of F.

Proof. Easy exercise. □

13.50 Exercise. Let F be a filter on X and x ∈ X. For each F ∈ F choose x_{F} ∈ F. Does the net (x_{F})_{F ∈F} (indexed by F directed by F′ ≥ F :⟺ F′ ⊆ F), also satisfy the equivalences of 13.49?

13.51 Remark: For a filter F to converge to x ∈ X, it suffices that F contains, for a fixed subbasis S of the topology, every Y ∈ S with x ∈ Y . Similarly, for a net to converge to x, it suffices that it is eventually in Y for every Y ∈ S with x ∈ Y .

Proof. Easy exercise. □

∗ ∗ ∗ Compactness ∗ ∗ ∗

13.52 Definition. Let X be a topological space and Y ⊆ X. An open cover of Y is a set C of open sets such that Y ⊆ ⋃_{O∈C} O. Y is compact if every open cover of Y admits a finite subcover, that is, there exist O_{1}, …, O_{n} ∈ C with Y ⊆ O_{1} ∪ … ∪ O_{n}.

Be aware that many authors require compact sets to be Hausdorff, and call our notion of compact “quasi-compact”.

13.53 Exercise. If X is compact and f: X → Y continuous, then f(X) is compact.

13.54 Theorem. Let X be a topoological space, and S a subbasis of the topology. The following are equivalent:

(1) X is compact, i.e., every open cover of X has a finite subcover.

(2) Every cover of X consisting of elements of S has a finite subcover.

(3) Every ultrafilter on X converges to some x ∈ X.

(4) Every filter on X has a cluster point x ∈ X.

Proof. (1 ⇒ 2) a fortiori.

(2 ⇒ 3) Suppose the ultrafilter U doesn’t converge. For every x ∈ X choose U_{x} ∈ S with x ∈ U_{x} and U_{x} ∉ U (possible by 13.51) and let A_{x} = X \ U_{x}. Then A_{x} ∈ U. Also, {U_{x} | x ∈ X} covers X so there exists a finite set Y ⊆ X with ⋃_{x∈Y} U_{x} = X. Therefore ∅ = ⋂_{x∈Y} A_{x} ∈ U, a contradiction.

(3 ⇒ 4) By 13.35, every filter F on X can be refined to an ultrafilter. This ultrafilter converges to some x ∈ X and then x is a cluster point of F by 13.44.

(4 ⇒ 1) Suppose C is an open cover of X that has no finite subcover. Then we may use {X \ O | O ∈ C} as base for a filter F on X. Let x ∈ X be a cluster point of F and O_{x} ∈ C with x ∈ O_{x}. Then F ∩ O_{x} ≠ ∅ for all F ∈ F. But (X \ O_{x}) ∈ F, a contradiction. □

13.55 Corollary. Let X be a topologial space. Let B ⊆ P(X) be a set of closed sets such that every closed subset of X is representable as an arbitrary intersection of finite unions of elements of B (or, equivalently, such that S = {(X \ A) | A ∈ B} is a subbasis of the topology on X). Then the following are equivalent.

(1) X is compact. (1’) For every set A ⊆ P(X) of closed subsets of X it is true that: if ⋂_{A∈A} A = ∅

then there exists a finite subset {A_{1}, …, A_{n}} ⊆ A with ⋂_{1≤i≤n} A_{i} = ∅. (2’) Like (1’), but restricted to sets of closed sets A ⊆ B. (3’) Every ultranet in X converges to some x ∈ X. (4’) Every net in X has a cluster point x ∈ X.

Proof. (1 ⇔ 1′) and (2 ⇔ 2′) by de Morgan. (3 ⇔ 3′) and (4 ⇔ 4′) by 13.48 and 13.49. □

13.56 Exercise. If X is Hausdorff, then we can separate disjoint compact sets C_{1}, C_{2} by open sets, i.e., there are open sets O_{1}, O_{2} such that C_{1} ⊆ O_{1}, C_{2} ⊆ O_{2} and O_{1} ∩ O_{2} = ∅. (First show that we can separate a compact set C from a point x ∈ X \ C by open sets.)

13.57 Exercise.

(i) Every closed subset of a compact space is compact. (ii) If X is Hausdorff, then every compact subset of X is closed.

∗ ∗ ∗ Product topology ∗ ∗ ∗

13.58 Definition. For i ∈ I (an arbitrary index set) let X_{i} be a topological space. Product topology on the cartesian product ∏_{i∈I} X_{i} is defined by a subbasis consisting of all sets of the form

S(j, O_{j}) = {(x_{i})_{i∈I} ∈ ∏_{i∈I} X_{i} | x_{j} ∈ O_{j}},

for some j ∈ I, and some O_{j} open ⊆ X_{j}. (Equivalently, the sets O_{j} could be restriced to members of a given basis or subbasis of the topology of X_{j}.)

Remark: the finer topology on ∏_{i∈I} X_{i} given by the basis

B = {∏_{i∈I} O_{i} | ∀i O_{i} open ⊆ X_{i}}

is called box topology.

Note that the projections p_{j}: (∏_{i∈I} X_{i}) → X_{j}, p_{j}((x_{i})_{i∈I}) = x_{j}, are continuous, both for product topology and for box topology.

13.59 Proposition. A net (x_{λ})_{λ∈Λ} in X = ∏_{i∈I} X_{i} converges to y = (y_{i})_{i∈I} in product topology, if and only if for every i ∈ I, its projection to X_{i}, (p_{i}(x_{λ}))_{λ∈Λ} converges to y_{i} in X_{i}.

Proof. Easy exercise. □

13.60 Theorem (Tychonoff). X = ∏_{i∈I} X_{i} (with product topology) is compact if and only if each X_{i} is compact.

Proof. Easy direction: if ∏_{i∈I} X_{i} is compact, then for each i, X_{i} is compact as the image of ∏_{i∈I} X_{i} under the projection onto the i-th coordinate, which is continuous. Conversely, to show compactness of X = ∏_{i∈I} X_{i}, consider an ultranet on X. The projection to the i-the coordinate is an ultranet on X_{i}, which converges, since X_{i} is compact. As all coordinates of the ultranet converge, the ultranet itself converges. □

We give another proof of Tychonoff’s theorem using a different criterion for compactness.

Proof. Easy direction: if ∏_{i∈I} X_{i} is compact and C is an open cover of X_{i}, then {S(i, O) | O ∈ C} is an open cover of X, which has a finite subcover S(i, O_{1}),…, S(i, O_{n}). Clearly, O_{1},…, O_{n} cover X_{i} and constitute a finite subcover of C.

Now assuming compactness of each X_{i}, to show compactness of X = ∏_{i∈I} X_{i}, consider a cover C by subbasis elements S(i, O). There must be some coordinate j ∈ I such that the open sets O occurring in sets S(j, O) ∈ C cover X_{j}. (Otherwise, by the axiom of choice, there would be a point (x_{i})_{i∈I} such that for all i, x_{i} is in none of the sets O with S(i, O) ∈ C, and therefore (x_{i})_{i∈I} is not covered by C.) As X_{j} is compact, there is a finite cover of X_{j} by open sets O_{1}, … O_{n} with S(j, O_{k}) ∈ C. Clearly then S(j, O_{1}), …, S(j, O_{n}) cover X. □