Topological space in which there are no close and compacts subsets (except for the empty set)

Question

Any example of those topological spaces? I cant think of no one :S

I think it must be infinite and it must not be T2, but no idea how to find one.

Answer 1

Consider $\Bbb R$ with the topology whse open sets are:

• $\emptyset$ and $\Bbb R$;
• the open halflines $(a,\infty)$ for $a\in\Bbb R$.

Then every closed set $(-\infty,a]$ admits a covering in open sets which admits no finite subcovering, namely $$ (-\infty,a]\subset\cup_{n=1}^\infty(a-n,\infty) $$