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!