I was browsing Gillman and Jerison for known relations between zero-sets, $C$-embedded sets and so on.

The spaces I'm considering are $T_{3.5}$.

There are two properties that pseudocompact spaces have

All countable $C$-embedded sets are compact

All countable zero-sets are compact

Where (1) is equivalent to pseudocompactness, but (2) isn't.

A countable closed discrete $C^*$-embedded $G_\delta$-set in a pseudocompact space doesn't have to be either a zero-set nor $C$-embedded, as Katetov's example $\mathbb{N}\subseteq\Lambda = \beta\mathbb{R}\setminus (\beta\mathbb{N}\setminus\mathbb{N})$ shows.

A zero-set need not be $C^*$-embedded, nor does a $C$-embedded set need to be closed.

A zero-set is $C^*$-embedded iff $C$-embedded.

However, a countable $C$-embedded set is closed.

Must a countable zero-set be $C$-embedded?