This is a cross post from [MSE](https://math.stackexchange.com/questions/4004859/characterization-of-pretty-compact-spaces).


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–Čech compactification coincides with its one point compactification.


An example of almost compact space is $[0,\omega\_1)$ for the first uncountable ordinal $\omega\_1$. All almost compact spaces are locally compact and pseudocompact.


**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 know that Stonean spaces are pretty compact. By a result of van Douwen, Kunen and van Mill ([There can be $C^\*$-embedded dense proper subspaces in $\beta\omega - \omega$](https://staff.fnwi.uva.nl/j.vanmill/papers/papers1989/canbe.pdf)) $\beta\mathbb{N}\setminus \mathbb{N}$ is consistently pretty compact. What are other examples of pretty compact spaces? Does there exist any characterization of pretty compact spaces or at least a strong necessary condition?



Characterization of pretty compact spaces
A partial answer: other examples of pretty compact spaces are uncountable powers of $\{0,1\}$ and $[0,1]$, and in general products of uncountably many non-trivial compact Hausdorff spaces. See Problem 3.12.24(c) in Engelking's *General Topology*, or [Glicksberg, *Stone-Čech compactifications of products*](https://doi.org/10.1090/S0002-9947-1959-0105667-4). If $a$ is in the product take $b$ in the product that differs everywhere from $a$. Then $\Sigma(b)$ is a subset of $X\setminus\{a\}$. As the product is $\beta\Sigma(b)$ it is also $\beta(X\setminus\{a\})$ (general result: if $X\subseteq Y\subseteq\beta X$ then $\beta X=\beta Y$).


As noted in the question extremally disconnected compact spaces have the same property; as these are quite different from product spaces finding a characterization that is not a direct translation seems difficult.


