T_2 and compactness

Any compact T_2 (= Hausdorff) space (X, \tau)  is on the brink of being compact and being T_2:

(1) If we endow X with a topology \tau' distinct from \tau such that \tau' \supseteq \tau, then (X, \tau') is not compact any more; and
(2) if we endow X with a topology \tau' distinct from \tau such that \tau' \subseteq \tau, we lose the the Hausdorff property.

More concisely, these statements say: compact T_2 spaces are maximal compact and minimal T_2. They are both quite straightforward to prove.

Which leads to the question about the converse of these statements, i.e.

(1) Is every maximal compact space T_2?
(2) Is every minimal T_2 space compact?

As for the first question, the answer is No. One example is the Alexandroff compactification \mathbb{Q}^* of the rationals with the Euclidean topology. Recall that the Alexandroff compactification of a Hausdorff space (X, \tau) is formed by setting X^* = X \cup \{\infty\} where \infty \notin X and endowing X^* with the topology generated by \tau \cup \{X^* \setminus F: F \textrm{ is compact in }X\}.

The key to showing that \mathbb{Q}^* is maximal compact is the following
Lemma. A space is maximal compact if and only if every compact subset is closed.

Moreover, we cannot separate any q \in \mathbb{Q} from \infty \in \mathbb{Q}^* by disjoint neighborhoods, so \mathbb{Q}^* is not Hausdorff.

The lemma above characterizes maximal compact spaces. On the other hand I lack a tool for tackling minimal Hausdorffness. So I can answer Question (1) above, but for its ”dual” question whether minimal T_2 implies compactness, I don’t have a clue. Any hints are appreciated!

About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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