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



metrizable neighborhoods of compact subsets
The answer to the first question is no. Let $K$ be the inner circle in the [Alexandroff's double circle](http://dantopology.wordpress.com/2012/10/13/alexandroff-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.


