# Answer to Question #15804 in Abstract Algebra for ran

Question #15804

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.

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.

Expert's answer

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 !

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 !

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