# Thread: A normal Moore space is completely normal

1. ## 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

2. 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 $\displaystyle F\subset \overline{G}\subset H$.