In my first attempt to prove that the Ultrafilter Lemma implies Alexander subbase theorem, I tried to prove that the Ultrafilter Lemma could be used to show that if $X$ is not compact, then there exists a maximal open cover for $X$ with no finite subcovers. After failing at it, I posted the question linked above, where I learnt an alternative way to do so.
However, this make me wonder about the different equivalences (in ZF) of the statement "every non compact space has a maximal open cover without finite subcovers".
It may be a bit silly, but I cannot figure it out if the above statement is equivalent to the Axiom of Choice or any of its restrictions.
So, I would appreciate any references or comments concerning this.