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Proof That The Class of All Real Valued Continuous Functions on a Completely Regular T1 Space Separates Points

Theorem

The class of of all real valued continuous functionson a completely regular t1 spaceseparates points.

Proof

Suppose we have a completely regular topological spaceand distinct pointsandofSinceis T1 the setis closed. Pointsandare distinct hence

The space (X,T) is completely regular hence a real valued continuous functiononexists such thatand

The function f separates a and b since

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