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This is a question about general topology:

Assume we are given a first countable Hausdorff space and a compact subset K.

Is it possible to find a countable basis of open neighborhoods of K ?

Usually, the idea in Hausdorff topology is that everything that is true for a point should also be true for a compact subset. But in this case, I am not sure how to construct the sequence of open neighborhoods out of the open neighborhoods of the points...

A related question is the following: Assume that the big space is not only first countable but locally metrizable. Is it then true that each compact subset has a metrizable neighborhood?

I am sure that the answers of these questions are in some textbooks about General Topology but so far I could not find anything.

Many thanks in advance, Tom

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The answer to the first question is no. Let $K$ be the inner circle in the Alexandroff's double circle. Any open neighborhood of $K$ is cofinite, so any base of neighborhoods of $K$ has size continuum. This counterexample has many nice additional properties: compact, hereditarily normal, union of two metrizable subspaces, among others. The answer is yes if you assume that your compact subset $K$ is countable, and that is an easy exercise.

The answer to the second question is yes if you assume regularity of the big space, because then you can cover $K$ with a finite number of open sets whose closures are metrizable. The union of these open sets will be a metrizable neighborhood of $K$. I´m not sure what happens without regularity.

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  • $\begingroup$ Thanks a lot for your answer and in particular, for the Alexandroff double circle link. I did not know that one so far. $\endgroup$ – Tom Sep 29 '14 at 16:39

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