If a space is first countable, does that imply it is locally compact?

Question

If $X$ is a first countable space can I then somehow show that $X$ is also locally compact? Or are there counterexamples?

Thanks for your time and best regards.

Answer 1

The rational numbers equipped with the subset topology from the reals is first countable but not locally compact (since every compact set will have empty interior, so it will not be a neighborhood of any point).