# 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 $H$ 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 $p$of 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.

If $H$ is an open set and there is a closed set $F\subset H$ then there is an open set $G$ such that $F\subset \overline{G}\subset H$.