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.