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Math 490 Notes 21

Local Compactness

A topological space (X, τ) is defined to be locally compact iff for each point x ∈ X, there is a compact set which contains a nbhd of x. If(X, τ) is T2, then it is locally compact iff each point x has a nbhd whose closure is compact.

If (X, τ) is compact, then it is also locally compact, since X is a set containing a nbhd of every point. Among our seven ”standard” topologies on R, all except τs and τcoc are locally compact.

Prop N21.1 Let (X, τ) be a locally compact space, A a closed subset of X. Then the sub- space (A, τA) is locally compact.

Prop N21.2 If (X, τ) is T2, then (X, τ) is locally compact iff, for each x ∈ X and for each nbhd U of x, there exists a nbhd V of x such that clτ V is compact and clτ V ⊆ U.

Prop N21.3 Let (X, τ) be locally compact and T2, A an open subset of X. Then (A, τA) is locally compact.

We now examine (briefly) the notion of ”compactification”. A homeomorphism between

(X, τ) and a subspace of (Y, µ) is called an topological embedding of (X, τ) into (Y, µ).

The existence of such an embedding makes (X, τ) a generalized subspace of (Y, µ). A dense topological embedding is one such that Ran f is dense in Y .

Def N21.1 ((Y, µ), ψ) is a compactification of a non-compact space (X, τ)if(Y, µ) is com- pact, and ψ :(X, τ) → (Y, µ) is a dense topological embedding. If, in addition, (Y, µ) is T2, then ((Y, µ), ψ) is called a T2 compactification of (X, τ). p. 2

It is common in mathematics to embed a space which lacks some important property in an ”enlarged” space which has the given property; the original space is then regarded as a

(generalized) subspace of the enlarged space.

Some simple compactifications can be described geometrically. If X is the open interval

(0, 1) in R, Y = [0, 1], and ψ : X → Y is the identity injection, then (Y, ψ) is a ”2-point compactification” of X. If Y ′ is the quotient space obtained by identifying 0 and 1 in Y , then (Y ′, ψ) is a ”1-point compactification” of X, and is homeomorphic to a circle in R2. If

S is the open unit square in R2 whose corner points are (0, 0), (0, 1), (1, 1) and (1, 0), an obvious compactification (K, ψ) is obtained by letting K = [0, 1] × [0, 1], with ψ : S → K again being the identity injection. If K′ is the quotient space obtained by identifying all point on the boundary of K with a single point, then (K′, ψ) is a 1-point compactification of S which is homeomorphic to the surface of a sphere in R3. Note that in each of these examples, the compactification is T2.

Of course most topological spaces do not have simple geometric descriptions. We now ex- amine two general ways of forming a 1-point compactification of any topological space.

Def N21.2 Let (X, τ) be a non-compact topological space. Let a ∈ X, and let Y = X ∪ {a}.

Let µ = τ ∪ {Y }. Then µ is a topology in which Y is the only nbhd of the point a; thus

(Y, µ) is compact because any open cover must cover a, and hence U = {Y } would be a

finite subcover of Y . Clearly X is dense in Y . Under the identity injection ψ, ((Y, µ), ψ) is a compactification of (X, τ), called the trivial 1-point compactification.

This compactification is of little interest in topology, and certainly does not preserve the T2 property. Of much greater interest is the following. p. 3

Def N21.3 Let (X, τ) be a non-compact, T2 topological space. Let a ∈ X, Y = X ∪ {a}, ¯ and let µ be the topology on Y with basis B = τ ∪ {Y − K ¯ K is τ-compact in X}. Because

(X, τ) is T2, τ-compact sets are τ-closed, and it follows that B is closed under finite inter- sections and covers Y , so B is a basis for a topology µ on Y . If we again let ψ : X → Y be the identity injection, then it can be shown that ((Y, µ), ψ) is a compactification of (X, τ), called the Alexandrov 1-point compactification.

The Alexandrov 1-point compactification does not, in general, preserve the T2 property, but:

Prop N21.4 If (X, τ) is a T2, non-compact topological space and ((Y, µ), ψ) is its Alexandrov

1-point compactification, then (Y, µ) is T2 iff (X, τ) is locally compact.

Proof : p. 4

The previous examples involving 1-point compactifications of (0, 1) and the unit square are both special cases of Alexandrov 1-point compactifications; note that in both cases, the orig- inal spaces were locally compact. The Alexandrov 1-point compactification of SΩ is denoted

SΩ, and is homeomorphic to the interval of ordinals [0, Ω] (recall that SΩ is the ordered set of all countable ordinals, and Ω is the smallest uncountable ordinal). Another commonly used example is the Alexandrov 1-point compactification of an infinite discrete space; in this case the nbhds of the compactification point are cofinite sets.

Some Additional Topological Properties

• A space is called Lindelof iff every open cover of the space has a countable subcover. Al- though Munkres does not regard Lindelof as a compactness property, it is a generalization of compactness, and clearly, (compact) ⇒ (Lindelof).

A T1 topological space is:

• T3 (or regular) iff, whenever A ⊆ X is closed and x ∈ A, there exist disjoint open sets U and V such that x ∈ U and A ⊆ V .

• T3.5 (or completely regular) iff whenever A ⊆ X is closed and x ∈ A, there exists

cont f :(X, τ) → [0, 1] such that f(x)=0 and f(A)= {1}.

• T4 (or normal) iff, whenever A, B are disjoint closed sets, there exist disjoint open sets

U, V such that A ⊆ U and B ⊆ V .

Prop N21.5 T4 ⇒ T3.5 ⇒ T3 ⇒ T2 ⇒ T1 ⇒ T0.

Proof : We’ve already proved T2 ⇒ T1 ⇒ T0, and we don’t have the tools to prove T4 ⇒ T3.5.

The other two implications (T3.5 ⇒ T3 ⇒ T2) are left as exercises.