If $A$ is subspace of topological space $X$ is compact and closure of $A$ is not compact then $X$ is particular point topology

Question

I am looking for a topological space $X$ which if $A\subset X$ is compact but closure of $A$ is not compact.

From this Find a topological space X and a compact subset A in X such that closure of A is not compact., I know that if $X$ is particular point topology then a closure of compact subspace of $X$ is not compact.

My question is

is there any topological space $X$, except particular point topology, such that a closure of compact subspace of $X$ is not compact? or is it true:

If $A$ is subspace of topological space $X$ is compact and closure of $A$ is not compact then $X$ is particular point topology?

Answer 1

Not in general. The discrete union (topological sum) of finitely many infinite spaces, each with the particular point topology, is another example: the set of particular points is finite, hence compact, but it's also dense, and the space is not compact. The product of two infinite spaces, each with the particular point topology, is yet another example.

The space described in this answer is a much less trivial example; the set $X\setminus P$ is compact, infinite, and dense in $X$.