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