# $\exists$ a non-compact space in which every proper open subset is compact?

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).

## 1 Answer

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.

## protected by RohanDec 6 '17 at 7:58

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?