A space X is called completely regular if and only if for any closed A X and any

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A space X is called completely regular if and only if for any closed A X and any point x epsilon XA there is a continuous function f : X rightarrow [0, 1] such that f(x) = 0 and A f-1(l) Prove that any completely regular space is regular. Note: A completely regular T1-space is usually called a Tychonoff space, but in the T-subscript tradition, such spaces are often called T3.5 or T31 / 2-spaces. This probably started as a joke, but it has become fairly standard. Some people have also suggested that such spaces be called T pi-spaces since they are "closer" to regular than they are to normal.

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