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Completely Regular Spaces

  1. Oct 29, 2005 #1
    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.
  2. jcsd
  3. Oct 29, 2005 #2


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    Let C be the complement of U?
  4. Oct 29, 2005 #3

    I feel stupid now

    I was thinking about this late at night, I'm not usually this dense.
    Last edited: Oct 29, 2005
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