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Prove: If the class *C* [X, R] of all real-valued continuous functions on a topological space X
separates points, then X is a Hausdorff space.

Proof. Let a, b ε X be distinct points. By hypothesis, there exists a continuous function f: X
R in *C* [X, R] such that f(a)
f(b). Now R is a Hausdorff space. Consequently there exist
disjoint open subsets G and H of R containing f(a) and f(b) respectively. The inverses f^{-1}[G] and
f^{-1}[H] are then disjoint, open and contain a and b respectively. Thus X is a Hausdorff space.

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