12.6. SMOOTH PARTITIONS OF UNITY 369

12.6 Smooth Partitions of UnityPartitions of unity were discussed earlier. Here the idea of a smooth partition of unity isconsidered. The earlier general result on metric space is Theorem 3.12.5 on Page 92. Recallthe following notation.

Notation 12.6.1 I will write φ ≺V to symbolize φ ∈Cc (V ) , φ has values in [0,1] , and φ

has compact support in V . I will write K ≺ φ ≺V for K compact and V open to symbolizeφ is 1 on K and φ has values in [0,1] with compact support contained in V .

Definition 12.6.2 A collection of sets H is called locally finite if for every x, thereexists r > 0 such that B(x,r) has nonempty intersection with only finitely many sets of H .Of course every finite collection of sets is locally finite. This is the case of most interest inthis book but the more general notion is interesting.

The thing about locally finite collection of sets is that the closure of their union equalsthe union of their closures. This is clearly true of a finite collection.

Lemma 12.6.3 Let H be a locally finite collection of sets of a normed vector space V .Then

∪H = ∪{

H : H ∈H}.

Proof: It is obvious⊇ holds in the above claim. It remains to go the other way. Supposethen that p is a limit point of ∪H and p /∈ ∪H . There exists r > 0 such that B(p,r) hasnonempty intersection with only finitely many sets of H say these are H1, · · · ,Hm. Then Iclaim p must be a limit point of one of these. If this is not so, there would exist r′ such that0 < r′ < r with B(p,r′) having empty intersection with each of these Hi. But then p wouldfail to be a limit point of ∪H . Therefore, p is contained in the right side. It is clear ∪His contained in the right side and so This proves the lemma. ■

A good example to consider is the rational numbers each being a set in R. This is not alocally finite collection of sets and you note that Q= R ̸= ∪{x : x ∈Q} . By contrast, Z isa locally finite collection of sets, the sets consisting of individual integers. The closure ofZ is equal to Z because Z has no limit points so it contains them all.

Lemma 12.6.4 Let K be a closed set in Rp and let {Vi}∞

i=1 be a locally finite sequenceof bounded open sets whose union contains K. Then there exist functions, ψ i ∈C∞

c (Vi) suchthat for all x ∈ K,1 = ∑

∞i=1 ψ i (x) and the function f (x) given by f (x) = ∑

∞i=1 ψ i (x) is

in C∞ (Rp) .

Proof: Let K1 = K \∪∞i=2Vi. Thus K1 is compact because it is a closed subset of a

bounded set and K1 ⊆V1. Let W1 be an open set having compact closure which satisfies

K1 ⊆W1 ⊆W 1 ⊆V1

Thus W1,V2, · · · covers K and W 1 ⊆ V1. Suppose W1, · · · ,Wr have been defined such thatWi ⊆Vi for each i, and W1, · · · ,Wr,Vr+1, · · · covers K. Then let

Kr+1 ≡ K \ ((∪∞

i=r+2Vi)∪(∪r

j=1Wj)).

It follows Kr+1 is compact because Kr+1 ⊆Vr+1. Let Wr+1 satisfy

Kr+1 ⊆Wr+1 ⊆W r+1 ⊆Vr+1, W r+1 is compact