422 CHAPTER 16. STONE’S THEOREM

Proof:1.)⇒ 2.)Let S be an open cover of S and let B be an open countably locally finite refinement

B= ∪∞n=1Bn

where Bn is an open refinement of S and Bn is locally finite. For B ∈Bn, let

En (B) = B\⋃k<n

(∪{B : B ∈Bk}).

Thus, in words, En (B) consists of points in B which are not in any set from any Bk fork < n.

Claim: {En (B) : n ∈ N, B ∈Bn} is locally finite.Proof of the claim: Let p ∈ S. Then p ∈ B0 ∈Bn for some n. Let V be open, p ∈ V,

and V intersects only finitely many sets of B1∪ ...∪Bn. Then consider B0∩V . If m > n,

(B0∩V )∩Em (B)⊆

[⋃k<m

(∪{B : B ∈Bk)

]C

⊆ BC0 .

In words, Em (B) has nothing in it from any of the Bk for k <m. In particular, it has nothingin it from B0. Thus (B0∩V )∩Em (B) = /0 for m > n. Thus p∈ B0∩V which intersects onlyfinitely many sets of S, no more than those intersected by V . This establishes the claim.

Claim: {En (B) : n ∈ N, B ∈Bn} covers S.Proof: Let p ∈ S and let n = min{k ∈ N : p ∈ B for some B ∈Bk}. Let p ∈ B ∈Bn.

Then p ∈ En (B).The two claims show that 1.)⇒ 2.).2.)⇒ 3.)Let S be an open cover and let

G ≡ {U : U is open and U ⊆V ∈S for some V ∈S}.

Then since S is regular, G covers S. (If p ∈ S, then p ∈U ⊆U ⊆V ∈S. ) By 2.), G has alocally finite refinement C, covering S. Consider

{E : E ∈ C}.

This collection of closed sets covers S and is locally finite because if p ∈ S, there existsV, p ∈ V, and V has nonempty intersections with only finitely many elements of C, sayE1, · · · ,En. If E ∩V ̸= /0, then E ∩V ̸= /0 and so V intersects only E1, · · ·,En. This shows2.)⇒ 3.).

3.)⇒ 4.) Here is a table of symbols with a short summary of their meaning.

Open covering Locally finite refinementS original covering B by 3. can be closed refinementF open intersectors C closed refinement