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I believe that the following problem have already been considered by some sophisticated topologist.

Definition 1. A non-compact Hausdorff topological space $X$ is called almost compact if its Stone-Cech compactification coincides with its one point compactification.

An example of almost compact space is $[0,\omega_1)$ for first uncountable ordinal $\omega_1$.

Definition 2. A compact Hausdorff space $X$ is called pretty compact if $X\setminus\{p\}$ is almost compact for all non-isolated points $p\in X$.

I would like to hear answers to any of the following questions.

Questions:

  • What are examples of pretty compact spaces?
  • Is it true that pretty compact spaces are extremally disconnected?
  • Is it true that pretty compact spaces contain dense extremally disconnected subspace?
  • Does there exist any characterization of pretty compact spaces?
Norbert
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  • I deleted my no longer relevant comments +1. – Tyrone Jan 29 '21 at 19:36
  • Are all almost compact spaces "large"? (I had to italicize because I confused myself rereading my sentence having all and almost right next to each other. Hah!) – Cameron Williams Jan 29 '21 at 20:02
  • @CameronWilliams, this is a soft question. In some sense they are large. Almost compact spaces are just 1 point behind their largest compactification. – Norbert Jan 29 '21 at 20:52
  • There are hardly any "almost compact" spaces, so I expect a dearth of examples. – Henno Brandsma Jan 30 '21 at 08:23
  • Indeed, the only two examples that I've found are from the book Pseudocompact topological spaces. M. Hrusak, A. Tamariz-Mascarua, M. Tkachenko. On the page 17 they say that $[0,\omega_1)$ is almost compact and Mrowka-Isbell space $\Psi(\mathcal{A})$ is almost compact for some specific maximal almost disjoint family in $2^\omega$. – Norbert Jan 30 '21 at 09:01
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    At least consistently, pretty compact spaces need not be extremally disconnted: By a result of van Douwen, Kunen and van Mill $\beta \Bbb N \setminus \Bbb N$ is consistently pretty compact, but, as it is well-known, not extremally disconnected. – Ulli Jan 30 '21 at 14:08
  • @Ulli, thank you for citing this paper! – Norbert Jan 30 '21 at 20:41

1 Answers1

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The question was partially answered on mathoverflow.net. It seems like there is no short clear description for such spaces since they include essentially different classes of topological spaces.

Norbert
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