Space Suggestion
Ostaszewski's space is a topology on $\omega_1$, with a somewhat involved construction by A. J. Ostaszewski in "On Countably Compact, Perfectly Normal Spaces" under the assumption of $\lozenge$ (a strengthening of the continuum hypothesis), which is perfectly normal and countably compact.
If ZFC is consistent, such a space cannot be proven to exist in ZFC alone, in fact, in ZFC + CH, and no such spaces in pi-base exist yet, but I assumed it would be admissible, considering the existence of S147 and S148. And I assumed the fact that it's construction is non-straightforward would be admissible, considering the existence of S179.
Rationale
This space appears in the aforementioned article, and it, as well as other spaces with similar properties, are mentioned in a variety of other texts in set-theoretical topology including zbMATH 0807.54002, 0978.54023 and 0862.54031.
It serves as a counterexample to some startling theorems that may hold when CH fails, such as "a locally compact, hereditarily normal space is hereditarily paracompact iff it does not include a copy of $\omega_1$" or "every locally countably compact, perfectly normal space is paracompact". As such, I think it's valuable to include in pi-base.
Of course, the implementation isn't that urgent. Finishing the implementation of #1712 and beginning that of #1808 are probably more important, but I didn't want to forget about it.
Relationship to other spaces and properties
In the originally cited paper, it is shown that it is locally countable, perfectly normal, hereditarily separable, non-compact, strongly zero-dimensional, locally compact, and countably compact.
For some more "concrete"(?) properties: by construction, it has cardinality $\aleph_1$. It can also be easily checked to be $T_0$. It has an isolated point, namely $0$.
Space Suggestion
Ostaszewski's space is a topology on$\omega_1$ , with a somewhat involved construction by A. J. Ostaszewski in "On Countably Compact, Perfectly Normal Spaces" under the assumption of $\lozenge$ (a strengthening of the continuum hypothesis), which is perfectly normal and countably compact.
If ZFC is consistent, such a space cannot be proven to exist in ZFC alone, in fact, in ZFC + CH, and no such spaces in pi-base exist yet, but I assumed it would be admissible, considering the existence of S147 and S148. And I assumed the fact that it's construction is non-straightforward would be admissible, considering the existence of S179.
Rationale
This space appears in the aforementioned article, and it, as well as other spaces with similar properties, are mentioned in a variety of other texts in set-theoretical topology including zbMATH 0807.54002, 0978.54023 and 0862.54031.
It serves as a counterexample to some startling theorems that may hold when CH fails, such as "a locally compact, hereditarily normal space is hereditarily paracompact iff it does not include a copy of$\omega_1$ " or "every locally countably compact, perfectly normal space is paracompact". As such, I think it's valuable to include in pi-base.
Of course, the implementation isn't that urgent. Finishing the implementation of #1712 and beginning that of #1808 are probably more important, but I didn't want to forget about it.
Relationship to other spaces and properties
In the originally cited paper, it is shown that it is locally countable, perfectly normal, hereditarily separable, non-compact, strongly zero-dimensional, locally compact, and countably compact.
For some more "concrete"(?) properties: by construction, it has cardinality$\aleph_1$ . It can also be easily checked to be $T_0$ . It has an isolated point, namely $0$ .