19.1. GENERAL TOPOLOGICAL SPACES 505

Proof: Since X is locally compact, there exists a basis of open sets whose closures arecompact U . Denote by C the set of all U ∈U which contain k and let C ′ denote the set ofall closures of these sets of C intersected with the closed set VC. Thus C ′ is a collectionof compact sets. There are finitely many of the sets of C ′ which have empty intersection.If not, then C ′ has the finite intersection property and so there exists a point p in all ofthem. Since X is a Hausdorff space, there exist disjoint basic open sets from U , A,B suchthat k ∈ A and p ∈ B. Therefore, p /∈ A contrary to the above requirement that p be in allsuch sets. It follows there are sets A1, · · · ,Am in C such that VC ∩A1 ∩ ·· · ∩Am = /0.LetUk ≡ A1∩·· ·∩Am. Then Uk ⊆ A1∩·· ·∩Am and so it has empty intersection with VC. Thusit is contained in V . Also Uk is a closed subset of the compact set A1 so it is compact andk ∈Uk.

For the second part, consider all such Uk. Since K is compact, there are finitely manywhich cover K Uk1 , · · · ,Ukn . Then let U ≡∪n

i=1Uki . It follows that U = ∪ni=1Uki and each of

these is compact so this set works. ■The following is Urysohn’s lemma for locally compact Hausdorff spaces.

Theorem 19.1.23 (Urysohn) Let (X ,τ) be locally compact and let H ⊆U whereH is compact and U is open. Then there exists g : X → [0,1] such that g is continuous,g(x) = 1 on H and g(x) = 0 if x /∈ V for some open set V such that V ⊆U such that V iscompact.

Proof: This involves using Lemma 19.1.22 repeatedly. First use this lemma to obtainV open such that its closure is compact and contained in U with V ⊇ H. Thus H ⊆ V ⊆V ⊆U,V compact.

Let D≡ {rn}∞n=1 be the rational numbers in (0,1). Using Lemma 19.1.22, let Vr1 be an

open set such thatH ⊆Vr1 ⊆V r1 ⊆V, V r1 is compact

Suppose Vr1 , · · · ,Vrk have been chosen and list the rational numbers r1, · · · ,rk in order,

rl1 < rl2 < · · ·< rlk for {l1, · · · , lk}= {1, · · · ,k}.

If rk+1 > rlk then letting p = rlk , let Vrk+1 satisfy

V p ⊆Vrk+1 ⊆V rk+1 ⊆V, V rk+1 compact

If rk+1 ∈ (rli ,rli+1), let p = rli and let q = rli+1 . Then let Vrk+1 satisfy

V p ⊆Vrk+1 ⊆V rk+1 ⊆Vq, V rk+1 compact

If rk+1 < rl1 , let p = rl1 and let Vrk+1 satisfy

H ⊆Vrk+1 ⊆V rk+1 ⊆Vp, V rk+1 compact

Thus there exist open sets Vr for each r ∈Q∩ (0,1) with the property that if r < s,

H ⊆Vr ⊆V r ⊆Vs ⊆V s ⊆V.

Now for D≡Q∩ (0,1) , in the following, t will be in D

f (x)≡min(inf{t ∈ D : x ∈Vt},1) , f (x)≡ 1 if x /∈⋃t∈D

Vt .