It is known that If $X$ is a Hausdorff space then every compact subspace of $X$ is closed. Hence closure of compact subspace of $X$ is also compact.
My question: is there any a $T_1$ space $X$ such that if $A$ is a compact subspace of $X$ then closure of $A$ is not compact?
If $X$ is a $T_1$ space, then every finite subset of $X$ is both compact and closed. However, it is still possible for $X$ to have an infinite compact subset whose closure is not compact; the set $X\setminus P$ in the space described in this answer is an example.