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

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
$$
(-\infty,a]\subset\cup_{n=1}^\infty(a-n,\infty)
$$
