268
IX – Multivariate Differential and Integral Calculus
If X is locally compact and A is compact, for every x ∈ A choose an open
neighbourhood V (x) of x, with compact closure V (x) ⊂ U ; By BL, A can be
covered by finitely many V (x i ); their union V is the answer to the question
since ¯
V =
V (x i ) is compact and contained in U .
Lemma 3. Let U 0 , . . . , U n be open sets and X their union. There exist open
sets V 0 , . . . , V n whose union is X and such that ¯
V i ⊂ U i for all i.
The V p are constructed by induction on p by requiring them to also satisfy
V 0 ∪ . . . ∪ V p ∪ U p+1 ∪ . . . ∪ U n = X .
As V 0 only needs to satisfy
X − (U 1 ∪ . . . ∪ U n ) ⊂ V 0 ⊂ ¯
V 0 ⊂ U 1 ,
it is obtained by applying lemma 2 to A = X − (U 1 ∪ . . . ∪ U n ) and U = U 0 .
Now if V 0 , . . . , V p−1 are constructed so that
X = U p ∪ V 0 ∪ . . . ∪ V p−1 ∪ U p+1 ∪ . . . ∪ U n ,
the V p are obtained by arguing in the same way about this new covering.
Lemma 4. Let X be a locally compact metric space, A a compact subset
of X and (U i ) 1≤i≤n a finite open covering of A. Then, there exist positive
valued continuous functions f i on X, with compact support and such that
Supp(f i ) ⊂ U i ,
f i (x) = 1 on A .
(17.4)
Set U 0 = X − A. Lemma 3 gives open sets V i (0 ≤ i ≤ n) covering X
and such that V i ⊂ V i ⊂ U i . For 0 ≤ i ≤ n, lemma 1 proves the existence of
continuous functions g i on X, with values in [0, 1], and such that
g i (x) = 1 if x ∈ V i , g i (x) = 0 if x ∈ X − U i .
(17.5)
Consider the function g =
g i . It has ≥ 0 values and is even > 0 on the V i ,
hence on their union X. The functions h i = g i /g(0 ≤ i ≤ n) are, therefore,
defined and continuous on X and satisfy
h i (x) = 1 for all x ∈ X ; but as
h 0 = g 0 /g vanishes on X − U 0 = A, the functions h i (1 ≤ i ≤ n) have sum 1
on A, each vanishing outside the corresponding open set U i . The h i still need
to be transformed into functions with compact support. But as A is compact,
lemma 2 shows the existence of an open set W with compact closure such
that
A ⊂ W ⊂ ¯
W ⊂ U 1 ∪ . . . ∪ U n
and lemma 1 that of a function p equal to 1 on A and vanishing outside W .
Multiplying the h i by p gives functions f i vanishing outside the U i and whose
IX – Multivariate Differential and Integral Calculus
If X is locally compact and A is compact, for every x ∈ A choose an open
neighbourhood V (x) of x, with compact closure V (x) ⊂ U ; By BL, A can be
covered by finitely many V (x i ); their union V is the answer to the question
since ¯
V =
V (x i ) is compact and contained in U .
Lemma 3. Let U 0 , . . . , U n be open sets and X their union. There exist open
sets V 0 , . . . , V n whose union is X and such that ¯
V i ⊂ U i for all i.
The V p are constructed by induction on p by requiring them to also satisfy
V 0 ∪ . . . ∪ V p ∪ U p+1 ∪ . . . ∪ U n = X .
As V 0 only needs to satisfy
X − (U 1 ∪ . . . ∪ U n ) ⊂ V 0 ⊂ ¯
V 0 ⊂ U 1 ,
it is obtained by applying lemma 2 to A = X − (U 1 ∪ . . . ∪ U n ) and U = U 0 .
Now if V 0 , . . . , V p−1 are constructed so that
X = U p ∪ V 0 ∪ . . . ∪ V p−1 ∪ U p+1 ∪ . . . ∪ U n ,
the V p are obtained by arguing in the same way about this new covering.
Lemma 4. Let X be a locally compact metric space, A a compact subset
of X and (U i ) 1≤i≤n a finite open covering of A. Then, there exist positive
valued continuous functions f i on X, with compact support and such that
Supp(f i ) ⊂ U i ,
f i (x) = 1 on A .
(17.4)
Set U 0 = X − A. Lemma 3 gives open sets V i (0 ≤ i ≤ n) covering X
and such that V i ⊂ V i ⊂ U i . For 0 ≤ i ≤ n, lemma 1 proves the existence of
continuous functions g i on X, with values in [0, 1], and such that
g i (x) = 1 if x ∈ V i , g i (x) = 0 if x ∈ X − U i .
(17.5)
Consider the function g =
g i . It has ≥ 0 values and is even > 0 on the V i ,
hence on their union X. The functions h i = g i /g(0 ≤ i ≤ n) are, therefore,
defined and continuous on X and satisfy
h i (x) = 1 for all x ∈ X ; but as
h 0 = g 0 /g vanishes on X − U 0 = A, the functions h i (1 ≤ i ≤ n) have sum 1
on A, each vanishing outside the corresponding open set U i . The h i still need
to be transformed into functions with compact support. But as A is compact,
lemma 2 shows the existence of an open set W with compact closure such
that
A ⊂ W ⊂ ¯
W ⊂ U 1 ∪ . . . ∪ U n
and lemma 1 that of a function p equal to 1 on A and vanishing outside W .
Multiplying the h i by p gives functions f i vanishing outside the U i and whose