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.
Best Answer
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).