Kenneth Kuttler

708 CHAPTER 25. STONE’S THEOREM AND PARTITIONS OF UNITY

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. ■

25.1 Partitions of Unity and Stone’s TheoremFirst recall that if S is nonempty, then dist(x,S) satisfies |dist(x,S)−dist(y,S)| ≤ d (x,y) .It was Lemma 3.12.1.

Theorem 25.1.1 Let S be a metric space and let S be any open cover of S. Thenthere exists a set F, an open refinement of S, and functions {φ F : F ∈ F} such that

φ F : S→ [0,1]

φ F is continuous

φ F (x) equals 0 for all but finitely many F ∈ F

∑{φ F (x) : F ∈ F}= 1 for all x ∈ S.

Each φ F is locally Lipschitz continuous which means that for each z there is an open set Wcontaining z for which, if x,y ∈W, then there is a constant K such that

|φ F (x)−φ F (y)| ≤ Kd (x,y)

708 CHAPTER 25. STONE’S THEOREM AND PARTITIONS OF UNITYThus 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)) > an? (5) =6,>0.It follows that the collection of open sets{E,(B): BEG} =B,is locally finite. In fact, B (p, 5) cannot intersect more than one of them. In addition tothis,S CU{E, (B):n EN, BEG}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. I25.1 Partitions of Unity and Stone’s TheoremFirst recall that if S is nonempty, then dist (x, 5) satisfies |dist (x, 5) — dist (y,S)| <d (x,y).It was Lemma 3.12.1.Theorem 25.1.1 Lez S be a metric space and let © be any open cover of S. Thenthere exists a set §, an open refinement of ©, and functions {9 : F € §} such thatop: S— [0,1@ - is continuous p (x) equals 0 for all but finitely many F € §V{Or (x): F €3}=I1forallxes.Each @ , is locally Lipschitz continuous which means that for each z there is an open set Wcontaining z for which, if x,y € W, then there is a constant K such thatld (x) — Op (Y)| S$ Kd (x,y)