Spaces where each compact subset has compact closure: have they already been studied?

Question

Let $(X, \tau) $ be a topological space be such that for every $K\subset X$ compact implies $\operatorname{cl}(K)$ is also compact.

I am interested in the study of such spaces [call $\textrm{G} $ space, if there is no name].

Let's start with few examples:

1. Hausdorff spaces: In a Hausdorff space a compact set is closed, hence it's a $\textrm{G}$ space.

2. KC space (A topological space in which compact sets are closed) gives a large class of examples of $\textrm{G}$ space.

All Hausdorff spaces are KC spaces and there are examples of non Hausdorff KC spaces [Co-countable topology on uncountable set].

3. Non KC but still $\textrm{G}$ space: Co-finite topology on an infinite set.

4. Non $\textrm{G}$ space : Particular point topology on an infinite set. Let $X$ be any set and $p\in X.$ Then define $\tau_p=\{U\subset X : p\in U\}\cup\{\emptyset\}.$ In $(X, \tau_p) $ the set $K=\{p\}$ is compact but $\operatorname{cl}(K) =X$ is not compact.

Hausdorff space $\subset $ KC space $\subset \textrm{ G}$ space

Is the study of such topological spaces already done?