Could someone help me to improve the proof writing?

Question

I will prove the following claim. I'm not a native English speaker. Could someone help me to improve the writing?

A regular pseudocompact Moore space is ccc and first countable.

Prove: I will cite a corollary to prove it. The corollary is this: A feebly compact, perfect space is ccc and first countable. Pseudocompactness implies feebly compactness; A Moore space is perfect. So a regular pseudocompact Moore space is ccc and first countable.

Thanks for your help.

Answer 1

First, there’s no need to mention regularity: by definition every Moore space is regular. There’s also no real reason to mention first countability, since it’s trivial that every developable space is first countable: anyone who knows what a Moore space is will also know that Moore spaces are first countable. I would replace pseudocompact, which is generally of interest only for Tikhonov spaces, with feebly compact. The lemma should be simply:

Lemma. Every feebly compact Moore space is ccc.

Proof. Since every Moore space is perfect, this is an immediate consequence of Proposition $2.3$ of Jack R. Porter & R. Grant Woods, ‘Feebly compact spaces, Martin’s axiom, and “diamond”’, Topology Proceedings $9$, Nr. $1$ ($1984$), pp. $105$-$121$: A feebly compact perfect space is ccc.