Examples of almost compact spaces.

Question

This question is a follow up question to this one

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

The only two examples of almost compact spaces I know are from the book Pseudocompact topological spaces. M. Hrusak, A. Tamariz-Mascarua, M. Tkachenko. On the page 17 authors 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 $\mathcal{A}\subset 2^\omega$.

I would like to know more on almost compact spaces, but I found almost nothing on this subject.

Questions:

• What are other examples of almost compact spaces?
• Is true that $\beta\mathbb{N}\setminus\{p\}$ is almost compact for $p\in\beta\mathbb{N}\setminus\mathbb{N}$ ?
• Is it true that $X\setminus \{p\}$ is almost compact whenever $X$ is extremally disconnected and $p\in X$.

Answer 1

Let $Y=\beta \Bbb N \setminus \{p\}$ for $p \in \Bbb N^\ast$. Then theorem 6.4/6.7 from Gillman and Jerrison tells us that $\beta Y=\beta \Bbb N$ and another standard theorem tells us that the one-point compactification of $Y$ also equals $\beta \Bbb N$. So $Y$ is almost compact.

8L in that book also shows that $\Omega$ (the square of $\omega_1 +1$ with $(\omega_1, \omega_1)$ removed), is another example of an almost compact space, as is the Tychonoff plank, which is closely related. They are introduced in excercise 10R as spaces with a unique uniformity and the non-compact ones are characterised as those Tychonoff $X$ that have $|\beta X\setminus X| =1$.

Answer 2

Concerning the third bullet: It should definitely read as "whenever X is extremally disconnected and compact". Otherwise, $\mathbb{N}$ is extremally disconnected, but $\mathbb{N} \setminus \{1\}$ is not almost compact. Furthermore, if $p$ is isolated in $X$, then $X \setminus \{p\}$ is compact, hence not almost compact as defined above. (Although, almost compactness should include compactness, which actually is, to my knowledge, the most common understanding.)

Anyway, if $X$ is extremally disconnected, compact and $p\in X$, $p$ not isolated in $X$, then $X \setminus \{p\}$ is almost compact:

We have to show, that $\beta (X \setminus \{p\}) = X$, i.e. that if $A, B \subset X \setminus \{p\}$ are completely separated in $X \setminus \{p\}$, then $A, B$ are completely separated in $X$:

There exist open, disjoint subsets $U, V$ of $X \setminus \{p\}$ such that $A \subset U$ and $B \subset V$. $U, V$ are also open in $X$. Since $X$ is extremally disconnected, the closures of $U, V$ are disjoint and compact, hence completely separated. Hence also $A, B$ are completely separated in $X$.