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Thread: 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 $\displaystyle H$ and $\displaystyle K$ are two separated sets of a normal Moore space $\displaystyle X$. Let $\displaystyle g_1,g_2,...g_n$ be the sequence of open covers of $\displaystyle X$. For each $\displaystyle n$ let $\displaystyle H_n$ denote the set of all points $\displaystyle p$of closure of $\displaystyle H$ such that no open set of $\displaystyle g_n$ which cotains $\displaystyle p$ contains a point of closure of $\displaystyle K$. 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 $\displaystyle H$ is an open set and there is a closed set $\displaystyle F\subset H$ then there is an open set $\displaystyle G$ such that $\displaystyle F\subset \overline{G}\subset H$.
    Last edited by Plato; Mar 19th 2011 at 03:16 PM.
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