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 and are two separated sets of a normal Moore space . Let be the sequence of open covers of . For each let denote the set of all points of closure of such that no open set of which cotains contains a point of closure of . Honestly, I don't know why this is possible as well as how can I proceed.

Please guide me.

Thaank you in advance