Is there non-compacted Hausdorff Space?

Question

If P is a topological property, then a space $(X, \tau)$ is said to be minimal $P$ (respectively, maximal) if $(X, \tau)$ has property $P$ but no topology on $X$ which is strictly smaller (respectively, strictly larger ) than τ has P.

A topological space is called KC space if every compact subset is closed.

Theorem:

1: Every minimal $KC$-topological space is compact.

2:Every maximal compact space is minimal $KC$ space.

3:Every Hausdorff space is $KC$ space.

Is every minimal $KC$-topological, maximal compact?

Is there non-compacted Hausdorff Space ?

Is there a Compact Space not to be Hausdorff?

Answer 1

An example of non-Hausdorff compact space is a compact neighborhood of the origins in the line with two origins. https://en.wikipedia.org/wiki/Non-Hausdorff_manifold

Answer 2

Any cofinite topology on an infinite set (like $\mathbb{N}$) is not Hausdorff (but $T_1$) and compact. The one-point compactification (aka Aleksandrov extension) of $\mathbb{Q}$ is KC, compact but not Hausdorff as well.

A reference to your first claim is here.

Do you have a reference for the second one too? The last one is classical.

A minimal KC space is compact, as we saw, and a KC compact space is maximal compact. So that answers your first question.