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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].

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

  2. 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?

Sourav Ghosh
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  • Each finite space is G. This gives a lot of non-T1 examples. (Note that KC spaces are T1.) – Ulli Mar 08 '23 at 17:40
  • @Ulli Both are obvious. A finite set is always compact but it is not closed unless the space is $T_1$ .For the second claim :${x}$ is compact $\implies$ closed in KC space $\implies$ $T_1$ – Sourav Ghosh Mar 09 '23 at 07:55
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    Well, I didn't say that it's not obvious. (Probably as obvious as that KC spaces are G.) But it indicates that you cannot expect too much nice properties within the class of G spaces. – Ulli Mar 09 '23 at 09:23
  • In example four, is this not just the discrete topology? so shouldn't ${x}$ also be closed and thus $cl({x})={x}$? sorry if I am missing something – Chris Apr 21 '23 at 16:44
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    @chris No. The topology $\tau_p$ is not discrete. It's known as particular point topology ( discrete iff $X$ is singleton) .Infact $\textrm{cl}({x})=\begin{cases} {x} &x\neq p\X&x=p\end{cases}$ – Sourav Ghosh Apr 21 '23 at 17:10
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    @QuitMSE thank you, for the clarification – Chris Apr 21 '23 at 18:44

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