I was recently asked by a student whether there exists a topological space which is not compact, but in which every proper open subset is compact. I haven't been able to give an example, or a proof that no such space exists. So far the best I've been able to show is that such a space cannot contain a closed compact subset (in particular, it cannot be T1).
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Consider $\Bbb N$, with proper open sets given by $U_n = \{x: x\le n\}$ and the empty set. Arbitrary unions of the $U_n$ are open, (either given by $U_{m}$ the maximum of the $n$ or by $\Bbb N$ if the $n$ are unbounded), as too are finite intersections.
Here every proper open set is finite, and thus trivially compact. However, it is easy to see that $\Bbb N$ itself is not compact with this topology, since it is covered by the collection of all proper open subsets, which admits no finite subcover.
