Space Suggestion: Dow plank

[Moniker1998]

Space Suggestion

~~First example of non-normal extremally disconnected space is $\mathbb{N}\cup \{p_A: A\in\mathcal{A}\}$, $\mathcal{A}$ maximal almost disjoint family, $p_A\in \text{cl}_{\beta\mathbb{N}} A\setminus\mathbb{N}$. It's the space witnessing that $\beta\mathbb{N}$ is not $T_5$ so name probably should reflect that.~~

Define a non-normal $P$-space as $X^2\setminus \{(\omega_2, \omega_2)\}$ where $X = \omega_2+1$ has topology letting $U\subseteq X$ be open iff its a $G_\delta$-set in the order topology. I would like to call it Dow plank since it's a space that appears in the article On F-spaces and F'-spaces by Dow, clearly analogous to something like the Tychonoff plank.

Relationship to other spaces and properties

It's an example of non-normal P-space.

[yhx-12243]

Is the first example the same as #1201?

[Moniker1998]

@yhx-12243 oh sorry I forgot that existed, since it was sitting there for so long. Not quite, but it's similar. And also showing that $\beta\mathbb{N}$ is non-normal. Moreover it's more specific