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## Main Question or Discussion Point

I have seen two different definitions of "completely regular" (one in my class and online, and the other in my textbook). I am having trouble seeing how these definitions are equivalent.

A space S is said to be completely regular if for every closed subset C of S and every point x in S-C there is a continuous function from S to I such that

f(x)=0 and f(C)=1

A space S is said to be completely regular if for every point p of S and for every open set U containing p, there is a continuous function of S into I such that f(p)=0 and f(x)=1 for all points x in S-U.

A space S is said to be completely regular if for every closed subset C of S and every point x in S-C there is a continuous function from S to I such that

f(x)=0 and f(C)=1

A space S is said to be completely regular if for every point p of S and for every open set U containing p, there is a continuous function of S into I such that f(p)=0 and f(x)=1 for all points x in S-U.