[Math] Topological space that is compact and $T_1$ but not Hausdorff (i.e. normal but not hausdorff)

Question

I was freshing up on some topology, and this text I'm reading mentions T1 does not imply Hausdorff. A few counter-examples are readily available, like the natural numbers under the co-finite topology.

But what if we place a restriction on the space to also be compact, the text doesn't mention anything about that and I can't come up with any examples of spaces that are compact and T1 but not Hausdorff. To state it otherwise, I'm looking for a space that is T1 but not normal(=compact and Hausdorff).

Answer 1

You've already given an example: the natural numbers (or any infinite set, really) under the co-finite topology. Given any open cover, fixing a single (non-empty) element of the cover yields an open set that has all but finitely-many of the natural numbers as elements. Thus, only finitely-many more elements of the cover are needed, forming a finite subcover.