If $X$ is a locally compact and metrizable space such that its Alexandroff compactification is not first countable. Does this imply that no other compactification of $X$ can be first countable? Why?
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There can be locally-compact metrizable spaces with non-first-countable Alexandroff compactifications but with other first-countable compactifications. This follows from a theorem of Banakh and Leiderman in "Uniform Eberlein compactifications of metrizable spaces" which states that a metrizable space $X$ has a first-countable uniform Eberlein compactification if and only if $|X|\leq\mathfrak{c}$. This implies that the discrete space on $\omega_1$ (which is trivially locally-compact and metrizable) has a first-countable compactification. But the Alexandroff compactification is not first-countable, since the open subsets of the "point at infinity" consist of all co-finite subsets of $\omega_1$ and any (sub-)basis must be uncountable. |
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