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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 $\beta X\setminus X$ ...

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....

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 ...

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 ...