# topological $KC$ space

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?

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

The discrete topology on $$X$$ is a maximal KC-topology.
• 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 ? – adin Apr 24 at 14:00
• @adin Yeah, a compact set in $\tau_2$ is still compact in $\tau_1$. Hence it's closed in $\tau_1$, hence closed in $\tau_2$. – YuiTo Cheng Apr 24 at 14:06