8.1. PARTITIONS OF UNITY 187

The following definition gives some notation.

Definition 8.1.2 If K is a compact subset of an open set, V , then K ≺ φ ≺V if

φ ∈Cc(V ), φ(K) = {1}, φ(X)⊆ [0,1],

where X denotes the whole metric space. Also for φ ∈Cc(X), K ≺ φ if

φ(X)⊆ [0,1] and φ(K) = 1.

and φ ≺V ifφ(X)⊆ [0,1] and spt(φ)⊆V.

Next is a useful theorem. Recall from Theorem 2.4.8, x→ dist(x,S) is continuous.

Theorem 8.1.3 Let H be a compact subset of an open set U in X where (X ,d) is ametric space in which the closures of balls are compact. Then there exists an open set Vsuch that

H ⊆V ⊆ V̄ ⊆U

with V̄ compact. There also exists ψ such that H ≺ ψ ≺ V , meaning that ψ = 1 on Hand spt(ψ) ⊆ V̄ . If U is an open subset of Rp, then there is an increasing sequence ofcontinuous functions ψn ∈Cc (U) such that limn→∞ ψn (x) = XU (x) .

Proof: Consider h→ dist(h,UC

). This continuous function achieves its minimum at

some h0 ∈ H because H is compact. Let δ ≡ 12 dist

(h0,UC

). The distance is positive

because UC is closed. Now H ⊆∪h∈HB(h,δ ) . Since H is compact, there are finitely manyof these balls which cover H. Say H ⊆ ∪k

i=1B(hi,δ ) ≡ V. Then, since there are finitelymany of these balls, let

V ≡ ∪ki=1B(hi,δ ),V ≡ ∪k

i=1B(hi,δ )

V is a compact set since it is a finite union of compact sets.To obtain ψ, let

ψ (x)≡dist(x,VC

)dist(x,VC)+dist(x,H)

Then ψ (x)≤ 1 and if x ∈ H, its distance to VC is positive and dist(x,H) = 0 so ψ (x) = 1.If x ∈ VC, then its distance to H is positive and so ψ (x) = 0. It is obviously continuousbecause the denominator is a continuous function and never vanishes since both VC and Hare closed so if either dist

(x,VC

)or dist(x,H) is 0, then x is in either VC or H. Thus, if

one of dist(x,VC

),dist(x,H) is 0, the other isn’t. Thus H ≺ ψ ≺V .

For the last claim, Let Cn ≡{

x ∈U : dist(x,UC

)≥ 1/n

}and let Hn ≡Cn∩B(0,n) for

n ∈ N. Then Hn is compact, the Hn are increasing in n, and ∪nHn = U . Now for some m,Hm ̸= /0, let Hm ≺ φ m ≺U from the first part Let ψ1 ≡ φ m. If ψ1, ...,ψn have been chosen,let ψn+1 = max

(ψ1, ...,ψn,φ n+1+m

). Then eventually, if x ∈ U, for all n large enough,

ψn (x) = 1 = XU (x) and if x /∈U, then all ψn (x) = 0. ■

Theorem 8.1.4 (Partition of unity) Let K be a compact subset of X and suppose

K ⊆V = ∪ni=1Vi, Vi open.

Then there exist ψ i ≺ Vi with ∑ni=1 ψ i(x) = 1 for all x ∈ K. If H is a compact subset of Vi

for some Vi, there exists a partition of unity such that ψ i (x) = 1 for all x ∈ H