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Math Help - A normal Moore space is completely normal

  1. #1
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    A normal Moore space is completely normal

    Hello;

    I face difficulties in understanding the proof of this theorem: Every normal Moore space is completely normal.

    First we suppose that H and K are two separated sets of a normal Moore space X. Let g_1,g_2,...g_n be the sequence of open covers of X. For each n let H_n denote the set of all points pof closure of H such that no open set of g_n which cotains p contains a point of closure of K. Honestly, I don't know why this is possible as well as how can I proceed.

    Please guide me.

    Thaank you in advance
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  2. #2
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    Edited reply.
    While taking my old dog on his afternoon walk, it pop into my brain what you are asking about. It has been nearly forty years since I thought about Moore’s Axiom 1. But that is what you have reference to. The axiom appears on page 1 of his 1932 Foundations of Point Set Topology. Although he developed the axiom some fifteen years before (even before Hausdorff). That axiom actually defines a developable space long before anyone even considered that concept.

    There is absolutely no way to get into a discussion of such a broad topic here. I suggest you get a copy of that book and read it.

    But here is what you are going for.
    If H is an open set and there is a closed set F\subset H then there is an open set G such that F\subset \overline{G}\subset H.
    Last edited by Plato; March 19th 2011 at 04:16 PM.
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