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Sorry for the somewhat vague title. My questions are as follows:

Suppose $T$ is a quasicompact space. Does there exist a finer topology on $T$ such that it becomes compact?

Suppose $T$ is a Hausdorff space. Does there exist a coarser topology on $T$ such that it becomes compact?

The way I see this is that quasicompact/Hausdorff is a upper/lower bound on how fine the topology can be. This bound is also tight, in the sense that if two topologies $\tau_1 \subset \tau_2$ are both compact, then $\tau_1 = \tau_2$. So the questions ask if we can always get inside this bound from some starting point $T$.

I am not sure what the answer is. I have tried using Zorn's lemma, but quasicompactness and Haustorff properties aren't preserved at the obvious upper/lower bound of chains of topologies on a space X (made from taking union/intersection of all topologies in the chain).

I have also tried to construct counterexamples, but they havn't really worked either. I suspect that my examples are all too "nice", but I feel there are few pointers as to what a counterexample would look like.

Thanks for taking the time to read this. I would greatly appreciate any insight into this question.

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    $\begingroup$ Do you need $T$ to remain Hausdorff for the second question? $\endgroup$
    – bitesizebo
    Aug 13 '20 at 14:25
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    $\begingroup$ @bitesizebo The use of quasicompact indicates that "compact = quasicompact and Hausdorff". $\endgroup$ Aug 13 '20 at 14:26
  • $\begingroup$ $\{\emptyset, T\}$ is compact but probably not what you are looking for.. This indeed suggests that you mean compact = open cover condition + Hausdorff. $\endgroup$ Aug 13 '20 at 15:10
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    $\begingroup$ I think this is the Bourbaki usage of "compact". $\endgroup$ Aug 13 '20 at 15:20
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Yes, counterexamples to both exist, they are examples of so-called minimal Hausdorff spaces and maximal (quasi)compact spaces (that are not compact Hausdorff). The introduction to this paper from 1971 shows it's already a classic topic of research, and gives references to such examples (by Ramanathan and Smythe /Wilkins among the first ones).

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  • $\begingroup$ This answers my question perfectly! Thank you! $\endgroup$ Aug 14 '20 at 11:52

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