Does a realcompact space have unique realcompactification it is $G_\delta$-dense in?

Question

In literature there exists the following definition:

$Y$ is a realcompactification of $X$ if $X$ is dense in $Y$ and $Y$ is realcompact.

What I actually want to define instead is this:

$Y$ is a realcompactification* of $X$ if $X$ is $G_\delta$-dense in $Y$ and $Y$ is realcompact.

Here $G_\delta$-dense means that for every non-empty $G_\delta$ subset $U\subseteq Y$ we have $U\cap X\neq \emptyset$.

For example, if $\nu X$ is the Hewitt realcompactification of $X$, then $X$ is $G_\delta$-dense in $\nu X$. More generally, if $\rho(\mathcal{Z})$ is the Wallman-Frink realcompactification of $X$ with respect to a strong delta normal base $\mathcal{Z}$ on $X$, then $\rho(\mathcal{Z})$ is $G_\delta$-dense in $X$.

Suppose that $Y$ is a realcompactification* of a realcompact space $X$. Is it always the case that $Y = X$? If $X$ is $z$-embedded in $Y$, then $Y = X$. This will hold for example if $X$ is Lindelof, or $Y$ is $T_6$, see Extensions of Zero-sets and of Real-valued Functions by Blair.

Answer 1

No, it can be $X\neq Y$.

Let $X$ be a discrete space with $|X| = \aleph_1$ and consider the one-point compactification $Y$ of $X$. If $y\in Y\setminus X$ and $U$ is a $G_\delta$-set containing $y$, then $U$ is a co-countable set. Thus $U\cap X\neq \emptyset$. This shows that $X$ is $G_\delta$-dense in $Y$.

Note that $Y$ cannot be a $T_6$ space, but if $Z\subseteq Y$ then $Z$ is a union of a compact space $Z\setminus X$ and a normal open (in $Z$) space $Z\cap X$, so that $Z$ is normal. It follows that $Y$ is a $T_5$ space.

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Here is a slight modification of this example, which is interesting on its own. Let again $X$ be a discrete space with $|X| = \aleph_1$. Let $\mathcal{Z} = \{A\subseteq X : |A| < \aleph_1\text{ or }|A^c|<\aleph_1\}$. Then $\mathcal{Z}$ is a strong delta normal base on $X$, and $\mathcal{F} = \{A\subseteq X : |A^c| < \aleph_1\}$ is a real $\mathcal{Z}$-ultrafilter on $X$ which isn't fixed. Then the Wallman-Frink realcompactification $Y = \rho(\mathcal{Z})$ of $X$ with respect to $\mathcal{Z}$ is such that $X$ is $G_\delta$-dense in $Y$, $X$ and $Y$ are realcompact, yet $Y\setminus X = \{\mathcal{F}\}$.

This example is interesting because its also an example of a strong delta normal base $\mathcal{Z}$ on a realcompact space $X$ and a real $\mathcal{Z}$-ultrafilter on $X$ that isn't fixed.

The space $Y$ is $T_5$, not compact, and not $T_6$.