Is every countably compact space feebly compact?

Question

A topological space is said to be feebly compact if every locally finite cover by nonempty open sets is finite.

Every compact space is feebly compact but how about countably compact spaces?

Answer 1

Each countably compact space is feebly compact. For the proof of this cliam suppose the opposite. Let $\mathcal U$ be a locally finite infinite open cover of a countably compact space $X$. For each $U\in\mathcal U$ choose a point $x_U\in U$. Then consider a cluser point $x$ of the set $\{x_U:U\in\mathcal U\}$.

@TXC, I recommend you to look at the Section 2 of my article "Pseudocompact paratopological groups that are topological" about different weak forms of compactness and relations between them.

Answer 2

In fact, every pseudocompact is always feebly compact. It can be reversed for Tychonoff spaces. This can be seen: A survey on star covering properties (Page 29).

Note that countable compactness implies pseudocompactness. A counterexample is the classical $\Psi$, which is Tychonoff and pseudocompact, but not countably compact.