Any compact (= Hausdorff) space is on the brink of being compact and being :
(1) If we endow with a topology distinct from such that , then is not compact any more; and
(2) if we endow with a topology distinct from such that , we lose the the Hausdorff property.
More concisely, these statements say: compact spaces are maximal compact and minimal . 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 ?
(2) Is every minimal space compact?
As for the first question, the answer is No. One example is the Alexandroff compactification of the rationals with the Euclidean topology. Recall that the Alexandroff compactification of a Hausdorff space is formed by setting where and endowing with the topology generated by .
The key to showing that 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 from by disjoint neighborhoods, so 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 implies compactness, I don’t have a clue. Any hints are appreciated!