Answer to Question #15804 in Abstract Algebra for ran

Question

1.Let X be a Hausdorff space. Let f:X→R be such that {(x,f(x)):x∈X} is a compact subset of X×R . Show that f is continuous.

2.Let X be a compact Hausdorff space. Assume that the vector space of real-valued continuous functions on X is finite dimensional. Show that X is finite.

Answer 1

1. Since {(x,f(x)):x∈X} is a compact subset of X×R then X is compact and f(X)
is compact. Thus if f is not continuous, then f(X)
cannot be continious, so
is have to be continuous.

2. C(X) is finitely dimensional, then
{e_1(x),...,e_n(x)} is base of C(X).
If X is infinite, then there are
infinite number of constant functions

f(x)=x, if
x=x_0
and
f(x)=0, if x <> x_0

All this functions form base
of C(X), but since all bases in finite dimensional space have the same
cardinality, then X have to be finite !