Space Suggestion: Realcompactification of first measurable cardinal with discrete topology

[Moniker1998]

Space Suggestion

Let $\nu (\kappa)$ be Hewitt realcompactification of $\kappa$ with discrete topology, where $\kappa$ is the first measurable cardinal.

Rationale

If $X$ is a Tychonoff extremally disconnected $P$-space with $|X| < \kappa$ then $X$ is discrete, yet if $X$ is discrete with $|X|\geq \kappa$ then $\nu X$ is a non-discrete extremally disconnected $P$-space. It makes sense to take the smallest example of this form.

Relationship to other spaces and properties

This space provides an example satifying the search https://topology.pi-base.org/spaces?q=tychonoff+%2B+extremally+disconnected+%2B+P-space+%2B+not+discrete

[Moniker1998]

We might as well add first measurable cardinal with discrete topology.

[prabau]

The existence of these spaces depends on some set-theoretic assumptions beyond ZFC. In the name of such spaces, we normally put the assumption as a prefix, for example (CH). What prefix should we use in this case?

[Moniker1998]

Another space of this type, which is essentially strong uniform topology on $\nu(\kappa)$ above.