Real and Complex Analysis (Rudin)

38 REAL AND COMPLEX ANALYSIS
(a) Characteristic functions of open sets are lower semicontinuous.
(b) Characteristic functions of closed sets are upper semicontinuous.
The following property is an almost immediate consequence of the definitions:
(c) The supremum of any collection of lower semicontinuous functions is lower
semicontinuous. The infimum of any collection of upper semicontinuous functions
is upper semicontinuous.
2.9 Definition The support of a complex function f on a topological space X
is the closure of the set
{x:f(x) :F o}.
The collection of all continuous complex functions on X whose support is
compact is denoted by Cc(X).
Observe that Cc(X) is a vector space. This is due to two facts:
(a) The support off + g lies in the union of the support off and the support
of g, and any finite union of compact sets is compact.
(b) The sum of two continuous complex functions is continuous, as are
scalar multiples of continuous functions.
(Statement and proof of Theorem 1.8 hold verbatim if "measurable function" is
replaced by "continuous function," "measurable space" by "topological space";
take clI(s, t) = s + t, or clI(s, t) = st, to prove that sums and products of continuous
functions are continuous.)
2.10 Theorem Let X and Y be topological spaces, and let f: X --+ Y be continuous.
If K is a compact subset of X, thenf(K) is compact.
PROOF If {v,,} is an open cover of f(K), then {f- 1( VJ} is an open cover of K,
hence K c:f- 1 (v,,1) U"· uf- 1 (V,.,,) for some (Xl' ... , (x,,, and therefore
f(K) c: v" 1 u ... u v"..
IIII
Corollary The range of any f e Cc(X) is a compact subset of the complex
plane.
In fact, if K is the support offe Cc(X), thenf(X) c:f(K) u {o}. If X is not
compact, then 0 e f(X), but 0 need not lie inf(K), as is seen by easy examples.
2.11 Notation In this chapter the following conventions will be used. The
notation
K-