25.1. PARTITIONS OF UNITY AND STONE’S THEOREM 709

Proof: By Stone’s theorem, there exists a locally finite open refinement F of S coveringS. For F ∈ F

gF (x)≡ dist(x,FC)

Letφ F (x)≡ (∑{gF (x) : F ∈ F})−1gF (x) .

Now∑{gF (x) : F ∈ F}

is a continuous function because if x ∈ S, then there exists an open set W with x ∈W andW has nonempty intersection with only finitely many sets of F ∈ F. Then for y ∈W,

∑{gF (y) : F ∈ F}=n

∑i=1

gFi (y).

Since F is a cover of S,∑{gF (x) : F ∈ F} ̸= 0

for any x ∈ S. Hence φ F is continuous. This also shows φ F (x) = 0 for all but finitely manyF ∈ F. It is obvious that

∑{φ F (x) : F ∈ F}= 1

from the definition.Let z ∈ S. Then there is an open set W containing z such that W has nonempty intersec-

tion with only finitely many F ∈F . Thus for y,x ∈W,∣∣∣φ Fj(x)−φ Fj

(y)∣∣∣≤ ∣∣∣∣gFj (x)∑

ni=1 gFi (y)−gFj (y)∑

ni=1 gFi (x)

∑ni=1 gFi (x)∑

ni=1 gFi (y)

∣∣∣∣If F is not one of these Fi, then gF (x) = φ F (x) = φ F (y) = gF (y) = 0. Thus there is nothingto show for these. It suffices to consider the ones above. Restricting W if necessary, we canassume that for x ∈W,

∑F

gF (x) =n

∑i=1

gFi (x)> δ > 0, gFj (x)< ∆ < ∞, j ≤ n

Then, simplifying the above, and letting x,y ∈W, for each j ≤ n,∣∣∣φ Fj(x)−φ Fj

(y)∣∣∣≤ 1

δ2

∣∣∣∣ gFj (x)∑F gF (y)−gFj (y)∑F gF (y)+gFj (y)∑F gF (y)−gFj (y)∑F gF (x)

∣∣∣∣≤ 1

δ2 ∆∣∣gFj (x)−gFj (y)

∣∣+ 1

δ2 ∆

n

∑i=1|gFi (y)−gFi (x)|

≤ ∆

δ2 d (x,y)+

∆

δ2 nd (x,y) = (n+1)

∆

δ2 d (x,y)

Thus on this set W containing z, all φ F are Lipschitz continuous with Lipschitz constant(n+1) ∆

δ2 . ■

The functions described above are called a partition of unity subordinate to the opencover S. A useful observation is contained in the following corollary.