Real and Complex Analysis (Rudin)

POSITIVE BOREL MEASURES 37
2.6 Theorem If {K,,} is a collection of compact subsets of a Hausdorff space
and if nIX K" = 0, then some finite subcollection of {K,,} also has empty intersection.
PROOF Put Y,. = K~. Fix a member Kl of {K,,}. Since no point of Kl belongs
to every K", {Y,.} is an open cover of K 1. Hence K 1 e y"1 U ... U Y,.n for
some finite collection {Y,.J This implies that
2.7 Theorem Suppose U is open in a locally compact Hausdorff space X,
K e U, and K is compact. Then there is an open set V with compact closure
such that
Ke Ve Ve U.
PROOF Since every point of K has a neighborhood with compact closure,
and since K is covered by the union of finitely many of these neighborhoods,
K lies in an open set G with compact closure. If U = X, take V = G.
Otherwise, let C be the complement of U. Theorem 2.5 shows tha!....!o
each pEe there corresponds an open set w" such that K e Wp and p ¢ Wp.
Hence {C (") G (") W p }, where p ranges over C, is a collection of compact sets
with empty intersection. By Theorem 2.6 there are points PI' ... , Pn E C such
that
IIII
The set
then has the required properties, since
V = G (") w,,1 (") ... (") W Pn
Ve G (") W PI (") ••• (") W pn •
IIII
2.8 Definition Let f be a real (or extended-real) function on a topological
space. If
{x:f(x) > IX}
is open for every reallX,fis said to be lower semicontinuous. If
{x:f(x) < IX}
is open for every reallX,fis said to be upper semicontinuous.
A real function is obviously continuous if and only if it is both upper and
lower semicontinuous.
The simplest examples of semicontinuity are furnished by characteristic functions: