This is a cross post from [MSE][1].

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 $\beta\omega - \omega$][2]) $\beta\mathbb{N}\setminus \mathbb{N}$ 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?


  [1]: https://math.stackexchange.com/questions/4004859/characterization-of-pretty-compact-spaces
  [2]: https://staff.fnwi.uva.nl/j.vanmill/papers/papers1989/canbe.pdf