The spaces are called $KC$-spaces in which every compact subset is closed.

let $(X,\tau_1)$ be a $KC$-topological space, is there a $KC$ topology space $(X, \tau_2)$, so that $ \tau_ 1 \subset \tau_2$?

If there is not this topology, can you give me an example?

  • 2
    $\begingroup$ Are you sure that $\tau_1=\tau_2$ is allowed? $\endgroup$ – ajotatxe Apr 24 at 13:21

The discrete topology on $X$ is a maximal KC-topology.

  • $\begingroup$ Excuse me, I can not see what you write.can you explain it? $\endgroup$ – adin Apr 24 at 13:40
  • $\begingroup$ @adin Discrete topology is Hausdorff, and every compact set is closed in Hausdorff spaces. It is also the finest topology on a space. $\endgroup$ – YuiTo Cheng Apr 24 at 13:41
  • $\begingroup$ Can I prove that "let a $(X,\tau_1)$ is a $KC$-topological space and $\tau_1 \subset \tau_2$, is space $(X, \tau_2)$ a $KC$-topological space ? $\endgroup$ – adin Apr 24 at 14:00
  • $\begingroup$ @adin Yeah, a compact set in $\tau_2$ is still compact in $\tau_1$. Hence it's closed in $\tau_1$, hence closed in $\tau_2$. $\endgroup$ – YuiTo Cheng Apr 24 at 14:06

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