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-Cech 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 $\beta\mathbb{N}\setminus \mathbb{N}$ 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?

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?