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?

1 is answered here. Please don't post the same question twice. — YuiTo Cheng, May 02 '19 at 12:34
Any finite space is compact, whether or not it is Hausdorff... $\Bbb R$ with the usual topology is Hausdorff but not compact.... If $X$ is a compact Hausdorff space then any strictly stronger topology on $X$ (if there is one) is Hausdorff but not compact.... I have not studied your 1st Q — DanielWainfleet, May 02 '19 at 18:04

Henno Brandsma · Answer 1 · 2019-05-04T08:37:13.030

1

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.

edited May 04 '19 at 08:37

answered May 02 '19 at 21:54

Henno Brandsma

250,824

Yes. KC+compact $\Rightarrow$ maximal compact and minimal KC $\Rightarrow$ KC+compact. Again, this question has already been asked – YuiTo Cheng May 02 '19 at 23:30
@YuiToCheng That's not the same as maximal compact $\implies$ minimal KC, is it? – Henno Brandsma May 03 '19 at 04:13
No, it's not the same. But the OP is asking whether minimal KC $\Rightarrow$ maximal compact. – YuiTo Cheng May 03 '19 at 04:22
It is quite easy to see that a KC compact space is both maximal compact and minimal KC. It is also quite easy to show that every maximal compact space is KC. The remaining implication that every minimal KC space is compact is the most difficult one and was proved relatively recently. – Adam Bartoš May 03 '19 at 12:02
@user87690 For the record, minimal KC $\Rightarrow$ compact is proved here – YuiTo Cheng May 05 '19 at 15:40
@YuiToCheng That’s the proof from the paper I quoted. – Henno Brandsma May 05 '19 at 21:07

score 0 · Answer 2 · answered May 02 '19 at 12:25

0

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

answered May 02 '19 at 12:25

Grey Fox

396

Is there non-compacted Hausdorff Space?

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