TITLE: Characterization of pretty compact spaces
QUESTION [7 upvotes]: This is a cross post from MSE.
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 $\beta\omega - \omega$) $\beta\mathbb{N}\setminus \mathbb{N}$ 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?

REPLY [9 votes]: 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. 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.