TITLE: Minimal Hausdorff
QUESTION [11 upvotes]: A Hausdorff space $(X,\tau)$ is said to be minimal Hausdorff if for each topology $\tau' \subseteq \tau$ with $\tau' \neq \tau$ the space $(X,\tau')$ is not Hausdorff.
Every compact Hausdorff space is minimal Hausdorff.
I would like to know:
1) Is every minimal Hausdorff space compact?
2) Does every Hausdorff topology contain a minimal Hausdorff topology?
Many thanks!

REPLY [12 votes]: The answer to both questions is no - see 7.5 in Porter and Woods book, "Extensions and Absolutes of Hausdorff Spaces", Springer-Verlag, 1988.  The space of rational numbers with the usual topology has no coarser minimal Hausdorff topology.  Every Hausdorff space can be embedded in a minimal Hausdorff space; in particular, if you start with a Hausdorff space X that is not Tychonoff and embed it in a minimal Hausdorff space Y, then Y can not be compact Hausdorff.