Is every compact subset of a second countable space closed?

Question

I know that this is true for Hausdorff spaces and metric spaces, which are Hausdorff spaces, but I can’t prove it for second countable spaces. Is it even true? Thanks!

Answer 1

No, it's not true. For the easiest example take a two point space where one of the points is open and the other is not. The point that is open is not closed, but it is compact because it is finite. Clearly every finite space is second countable.

Answer 2

$X=\Bbb N$ in the co-finite topology is $T_1$ but not Hausdorff and trivially second countable as there are only countably many open sets, and every subset of it is compact, but only the finite ones and $X$ are closed...