Kenneth Kuttler

424 CHAPTER 16. STONE’S THEOREM

Proof: Let S be an open cover. Well order S. For B ∈S,

Bn ≡ {x ∈ B : dist(x,BC)< 1

2n }, n = 1,2, · · · .

Thus Bn is contained in B but approximates it up to 2−n. Let

En (B) = Bn \∪{D : D≺ B and D ̸= B}

where ≺ denotes the well order. If B, D ∈S, then one is first in the well order. Let D≺ B.Then from the construction, En (B)⊆ DC and En (D) is further than 1/2n from DC. Hence,assuming neither set is empty,

dist(En (B) ,En (D))≥ 2−n

for all B, D ∈S. Fatten up En (B) as follows.

Ẽn (B)≡ ∪{B(x,8−n) : x ∈ En (B)}.

Thus Ẽn (B)⊆ B and

dist(

Ẽn (B), Ẽn (D))≥ 1

2n −2(

≡ δ n > 0.

It follows that the collection of open sets

{Ẽn (B) : B ∈S} ≡Bn

is locally finite. In fact, B(

p, δ n2

)cannot intersect more than one of them. In addition to

this,S⊆ ∪{Ẽn (B) : n ∈ N, B ∈S}

because if p ∈ S, let B be the first set in S to contain p. Then p ∈ En (B) for n large enoughbecause it will not be in anything deleted. Thus this is an open countably locally finiterefinement. Thus 1.) in the above theorem is satisfied.

16.1 Partitions Of Unity And Stone’s TheoremFirst observe that if S is a nonempty set, then dist(x,S) satisfies |dist(x,S)−dist(y,S)| ≤d (x,y) . To see this,

|dist(x,S)−dist(y,S)| ≤ d (x,y)

To see this, say dist(x,S) is the larger of the two. Then there exists z ∈ S such that

dist(y,S)≥ d (y,z)− ε

It follows that

|dist(x,S)−dist(y,S)|= dist(x,S)−dist(y,S)

424 CHAPTER 16. STONE’S THEOREMProof: Let G be an open cover. Well order G. For B € G,1B, = {x €B: dist (x,B°) < aah n=1,2,-::.Thus B,, is contained in B but approximates it up to 2~”. LetE, (B) = B, \U{D: D = B and D F B}where ~ denotes the well order. If B, D € G, then one is first in the well order. Let D < B.Then from the construction, E, (B) C D© and E,, (D) is further than 1/2” from D©. Hence,assuming neither set is empty,dist (E, (B),E,(D)) >2™”for all B, D € G. Fatten up E,, (B) as follows.——_E, (B) = U{B (x,8-") : x € E, (B)}.——_Thus E,, (B) C B anddist (En (B),En(D)) > =-2(5) =5,>0.It follows that the collection of open sets{E, (B): BEG} =8,is locally finite. In fact, B (p, 5) cannot intersect more than one of them. In addition tothis,S CU{E, (B):n EN, Be G}because if p € S, let B be the first set in G to contain p. Then p € E, (B) for n large enoughbecause it will not be in anything deleted. Thus this is an open countably locally finiterefinement. Thus 1.) in the above theorem is satisfied. J16.1 Partitions Of Unity And Stone’s TheoremFirst observe that if S is a nonempty set, then dist (x,S) satisfies |dist (x,S) — dist (y,S)| <d (x,y). To see this,Idist (x, 8) — dist (y,)| <d (x,y)To see this, say dist (x, S) is the larger of the two. Then there exists z € S such thatdist (y, S) 2d (y,z) TeIt follows that|dist (x, S) — dist (y,S)|= dist (x,S) —dist(y,S)