36.1. PARTITIONS OF UNITY 1255

Lemma 36.1.6 Let K be a closed set and let {Vi}∞

i=1 be a locally finite list of boundedopen sets whose union contains K. Then there exist functions, ψ i ∈C∞

c (Vi) such that for allx ∈ K,

1 =∞

∑i=1

ψ i (x)

and the function f (x) given by

f (x) =∞

∑i=1

ψ i (x)

is in C∞ (Rn) .

Proof: Let K1 = K \∪∞i=2Vi. Thus K1 is compact because K1 ⊆V1. Let

K1 ⊆W1 ⊆W 1 ⊆V1

Thus W1,V2, · · · ,Vn covers K and W 1 ⊆ V1. Suppose W1, · · · ,Wr have been defined suchthat Wi ⊆Vi for each i, and W1, · · · ,Wr,Vr+1, · · · ,Vn 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

Continuing this way defines a sequence of open sets, {Wi}∞

i=1 with the property

Wi ⊆Vi, K ⊆ ∪∞i=1Wi.

Note {Wi}∞

i=1 is locally finite because the original list, {Vi}∞

i=1 was locally finite. Now letUi be open sets which satisfy

W i ⊆Ui ⊆U i ⊆Vi.

Similarly, {Ui}∞

i=1 is locally finite.

Wi Ui Vi

Since the set, {Wi}∞

i=1 is locally finite, it follows ∪∞i=1Wi = ∪∞

i=1Wi and so it is possibleto define φ i and γ, infinitely differentiable functions having compact support such that

U i ≺ φ i ≺Vi, ∪∞i=1W i ≺ γ ≺ ∪∞

i=1Ui.

Now define

ψ i(x) ={

γ(x)φ i(x)/∑∞j=1 φ j(x) if ∑

∞j=1 φ j(x) ̸= 0,

0 if ∑∞j=1 φ j(x) = 0.