When is $X$ $P$-embedded in $\beta X$?

Question

Let $X$ be a space and $A\subseteq X$ a subspace. Recall that $A\subseteq X$ has the extension property with respect to a space $Z$ if every continuous map $A\rightarrow Z$ extends over all of $X$. We say that:

1. $A$ is $C^*$-embedded in $X$ if it has the extension property with respect to $[0,1]$.
2. $A$ is $C$-embedded in $X$ if it has the extension property with respect to $\mathbb{R}$.
3. $A$ is $P^\kappa$-embedded in $X$ for some infinite cardinal $\kappa$ if it has the extension property with respect to every Banach space $E$ of weight $\leq\kappa$.
4. $A$ is $P$-embedded in $X$ if it has the extension property with respect to every Banach space $E$.

Then $P$-embeddded $\Rightarrow$ $P^\omega$-embeddded $\Leftrightarrow$ $C$-embedded $\Rightarrow$ $C^*$-embedded. In general only the one indicated arrow is reversible.

Here are some examples. A space $X$ is normal if and only if each closed $A\subseteq X$ is $C$-embedded if and only if each closed $A\subseteq X$ is $C^*$-embedded (this is the Tietze extension theorem). Every closed subspace of a paracompact space is $P$-embedded, while in general a space $X$ is collectionwise normal if and only if each closed $A\subseteq X$ is $P$-embedded.

More to the point, every Tychonoff space $X$ is densely $C^*$-embedded in its Stone-Cech compactification $\beta X$, and in turn this property characterises the Stone-Cech compactification. On the other hand it is possible for a Tychonoff space $X$ to be $C$-embedded in its Stone-Cech compactification. In fact this will occur if and only if $X$ is pseudocompact.

This brings me to my main question.

Which Tychonoff spaces are $P$-embedded in their Stone-Cech compactifications? Are these already the compact spaces? As remarked above, a pseudocompact Tychonoff space $X$ is $P^\omega$-embedded in $\beta X$. What can be said in general about the Tychonoff spaces which are $P^\kappa$-embedded in their Stone-Cech compactifications for any given $\kappa>\omega$?

According to Hewitt a space $X$ is pseudocompact if and only if it is $G_\delta$-dense in $\beta X$. My guess would be that if $X$ is $P^\kappa$-embedded in $\beta X$, then it should be $G_\kappa$-dense in $\beta X$, and hence if $X$ is $P$-embedded in $\beta X$, then it should already equal $\beta X$ (i.e. be compact).

Answer 1

Theorem. A Tychonoff space $X$ is $P$-embedded in its Stone-Čech compactification iff $X$ is pseudocompact.

Proof. $(\Rightarrow)$ If $X$ is $P$-embedded in $\beta X$ then $X$ is $C$-embedded in $\beta X$, so $X$ is pseudocompact.

$(\Leftarrow)$ Let $X$ be any pseudocompact space and $f$ be any continuous map from $X$ to a metrizable (in particular, Banach) space $E$. Since $f(X)$ is a pseudocompact subspace of a metrizable space, $f(X)$ is compact, see an implication ii$\Rightarrow$ i here. Since $f(X)$ is Tychonoff space, there exists a homeomorphic embedding $g$ of $f(X)$ into a Tychonoff cube $I^A$, where $I$ is the segment $[0,1]$ endowed with the usual topology. For each $\alpha\in A$ let $\pi_\alpha:I^A\to I$ be the projection onto the $\alpha$-th factor. A map $\pi_\alpha gf:X\to I$ has a continuous extension $h_\alpha$ to $\beta X$. It is easy to show that a diagonal $h:\Delta_{\alpha\in A} h_\alpha:\beta X\to I^A$, $x\mapsto (h_ \alpha(x))_{\alpha\in A}$ is a continuous map (see, for instance, Proposition 2.3.6 form [Eng]). It is easy to check that $h(x)=fg(x)$ for each point $x\in X$. Since the space $f(X)$ is compact, a $gf(X)$ is compact too, so $gf(X)$ is a closed subset of $I^A$. Since $gf(X)$ is a dense subset of a compact set $h(\beta X)$, $gf(X)=h(\beta X)$. Then $g^{-1}h:\beta X\to f(X)\subset E$ is a required continuous extension of the map $f$ from $X$ to $\beta X$. $\square$

References

[Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.