# Countable zero-sets are $C$-embedded?

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

1. All countable $$C$$-embedded sets are compact

2. 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?

Most of the following can be found in Gillman and Jerison.

Lemma Subsets $$A,B\subseteq X$$ are completely separated if and only if they are contained in disjoint zero-sets. $$\quad\blacksquare$$

A subset $$A\subseteq X$$ is well-embedded if it is completely separated from any disjoint zero-set.

Corollary Every zero-set is well-embedded. $$\quad\blacksquare$$

A subspace $$A\subseteq X$$ is z-embedded if every zero-set of $$A$$ is the trace of a zero-set of $$X$$. Every C$$^*$$-embedded subspace is z-embedded. The following result is found in the paper On the Structure of a Class of Archimedian Lattice-Ordered Algebras by Henriksen and Johnson, where it is attributed to Jerison.

Proposition A Lindelöf subspace of a Tychonoff space is z-embedded. $$\quad\blacksquare$$

In fact, there is a kind of converse for this last statement: a Tychonoff space $$X$$ is z-embedded in every Tychonoff space containing it if and only if $$X$$ is either Lindelöf or almost-compact.

Finally we have the following, which I believe is due to Blair and Hager.

Proposition A subspace $$A\subseteq X$$ is C-embedded if and only if it is both z-embedded and well-embedded.

This gives us the following.

Corollary Any z-embedded zero set is C-embedded. $$\quad\blacksquare$$

Corollary If $$X$$ is a Tychonoff space and $$A\subseteq X$$ is a Lindelöf zero-set, then $$A$$ is C-embedded in $$X$$. $$\quad\blacksquare$$

Corollary If $$X$$ is a Tychonoff space, then every countable zero-set of $$X$$ is C-embedded. $$\quad\blacksquare$$

• I think the paper Extensions of Zero-Sets and of Real-Valued Functions by Blair and Hager collects a few statements and has a decent bibliography. A lot of papers cite the text book Normal Topological Spaces by Alò and Shapiro as general reference for z-embeddings, but the book seems to be pretty rare and I've never seen a copy myself. Commented Jul 1 at 11:26
• I never realised before that Blair didn't use the term, as it's fairly standard now. I guess I must have picked it by reading K. Yamazaki's papers. Apparently the terminology was introduced in Moran's 1970 paper Measures on Metacompact Spaces. Commented Jul 1 at 13:03
• Perhaps its interesting to point out that corollary 4.2 from Extensions of Zero-Sets and of Real-Valued Functions by Blair and Hager also gives us a partial converse i.e. a Lindelöf C-embedded set is closed. This is a particular case of the fact that Lindelöf spaces are realcompact and a realcompact $C$-embedded set is closed. Commented Jul 11 at 13:31
• Or another: a Lindelöf subspace of a Tychonoff space is a zero-set if and only if it is a C-embedded $G_\delta$-subset. Commented Jul 11 at 15:04
• Call $S$ strongly well-embedded if $S$ is completely separated from each disjoint closed set. This property implies $S$ is closed, and a zero-set if its $G_\delta$. A space $X$ is normal iff every closed subspace is strongly well-embedded. And a Lindelöf well-embedded subspace is strongly well-embedded. One of the implications in above comment follows from these facts. Commented Jul 13 at 12:45

All countable zero-sets are $$C$$-embedded.

I'll need the following from the book "Rings of continuous functions" by Gillman and Jerison.

Completely separated: Sets $$D_1, D_2$$ are completely separated if there exists disjoint zero sets $$D_1\subseteq Z_1, D_2\subseteq Z_2$$.

Theorem 1.17: If $$Y\subseteq X$$, then $$Y$$ is $$C^*$$-embedded in $$X$$ iff completely separated sets in $$Y$$ are completely separated in $$X$$.

Exercise 3B.1: If $$D$$ is countable and $$F$$ is closed, $$D\cap F = \emptyset$$, then there exists a zero-set $$Z$$ with $$D\cap Z = \emptyset$$ and $$Z\supseteq F$$.

Exercise 1F.1: A $$C^*$$-embedded zero-set is $$C$$-embedded

Let $$D$$ be a countable zero-set and $$D_1, D_2\subseteq D$$ be disjoint zero-sets of $$D$$. Thus $$D_1, D_2$$ are disjoint countable closed sets in $$X$$. From exercise 3B.1 there exists a zero set $$Z_1\supseteq D_1$$ of $$X$$ with $$Z_1\cap D_2 = \emptyset$$, and we can assume $$Z_1\subseteq Z$$ by replacing $$Z_1$$ with $$Z_1\cap Z$$ since $$Z$$ is a zero-set. Similarly, since $$Z_1$$ is countable, we can now find a zero-set $$Z_2$$ with $$Z_1\cap Z_2 = \emptyset$$ and $$Z_2\supseteq D_2$$. Thus $$D_1, D_2$$ are completely separated in $$X$$. From Urysohn's lemma 1.17, $$D$$ is $$C^*$$-embedded, hence $$C$$-embedded.