# If every compact set is closed, then is the space Hausdorff?

I know that in a Hausdorff space, every compact set is closed.

However, is it true that if every compact set is closed, then the space is necessarily Hausdorff?

• I suspect that you meant to ask the question in the title, not the one in the body of the question. – Brian M. Scott Mar 12 '13 at 19:26
• The title and the body ask different questions. – Asaf Karagila Mar 12 '13 at 19:26
• The title and question are different. The title asks if every compact set is also a closed set, is the space Hausdorff. The question asks if every set is closed and compact, is it Hausdorff. Which do you mean? – anonymous Mar 12 '13 at 19:26
• Assuming the version in the title, such a space is at least $T_1$ because points are closed. – JSchlather Mar 12 '13 at 19:30

I’m answering the question in the title. Let $X$ be an uncountable set, and let $\tau$ be the co-countable topology on $X$. The compact sets in $\langle X,\tau\rangle$ are precisely the finite sets, which are all closed, but $X$ is not Hausdorff.