All Questions

Recently I've proven the following theorem Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent: Every compactification of $X$ is zero-dimensional....

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 ...

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets, and a topological space, $X$ is called completely regular exactly in case points can be ...

Question: Let $X$ be a continuum and $p \in X$. Under what conditions does there exist a compactification $\gamma (X-p)$ with $\gamma (X-p) - (X-p)$ connected and nondegenerate? Throughout, $X$ is a ...

Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications: Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...

Probably $\beta \mathbb N$ is not an absolute retract (is there an easy argument for this?), but I'd be interested to know what happens in the class of extremally disconnected (compact) spaces. Is it ...