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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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    The only extra info on $X$ you have is that it is not separable. Any metrizable $X$ has a metrizable compactification iff it is separable (for $X$ locally compact the Aleksandrov one will do, non-separable spaces will have others). So the question comes down to: does a non-separable metrizable space have a first countable compactification?2010-12-03

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