Regular non-normal extremally disconnected space

[Moniker1998]

The space (or spaces) $\Pi$ from Gillman and Jerison is an example of Tychonoff extremally disconnected space which isn't normal.
The construction is similar to Mrówka-Isbell space but the construction is based on $\beta\mathbb{N}$.

Note: In pull request #357 I mentioned that [extremally disconnected + regular => zero-dimensional] so all extremally disconnected $T_3$ spaces are $T_{3.5}$.

[prabau]

@Moniker1998 FYI, notice how I edited the description above to mention the pull request number. That automatically creates a link back from the other issue/PR to this one. Very convenient to navigate between the two.

[Moniker1998]

Any ideas for how this space could be named on pi-base?

It's related to Mrówka-Isbell spaces, such as https://topology.pi-base.org/spaces/S000057

Here's the construction. Fix a sequence of rationals converging to each irrational, say $\mathcal{A} = \{ E_x : x\in\mathbb{R}\setminus \mathbb{Q} \}$ is the family of such sequences. This step is the same as for S57. Equip $\mathbb{Q}$ with discrete topology (this corresponds to making all points in $\mathbb{Q}$ isolated). For each $x\in\mathbb{R}\setminus\mathbb{Q}$ choose some $p_x\in \text{cl}_{\beta\mathbb{Q}} E_x \setminus \mathbb{Q}$. Then $\Pi = \mathbb{Q}\cup \{p_x : x\in\mathbb{R}\setminus \mathbb{Q}\}\subseteq \beta\mathbb{Q}$.

Important note: Here $\beta\mathbb{Q}$ denotes the Stone-Cech compactification for $\mathbb{Q}$ equipped with discrete topology (!). Thus it's homeomorphic to $\beta\mathbb{N}$.

(EDIT by @StevenClontz to make set braces appear [looks like \\{ is necessary].)

[Moniker1998]

Gillman and Jerison contribute this space to Katětov, so we could label it as Katětov's rational sequence topology space?

[Moniker1998]

Some properties of $\Pi$:

1. Every subspace of $\Pi$ is extremally disconnected
2. Every subspace of $\Pi$ is $G_\delta$
3. $\Pi$ is not first countable
4. $\Pi$ is not normal
5. $\Pi$ is Tychonoff
6. $\Pi$ is not pseudocompact (Mrówka-Isbell space is pseudocompact iff it's constructed from a maximal ADF. In contrast, one of the exercises in Gillman and Jerison shows that even if we construct $\Pi$ using a maximal ADF, it won't be pseudocompact)
7. $\Pi$ is realcompact