TITLE: Alexandroff compactification question
QUESTION [9 upvotes]: 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?

REPLY [6 votes]: 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.