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!



Minimal Hausdorff
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.


