What if every nontrivial closed subset is compact?
111360
Topological spaces in which every proper closed subset is compact
92005
Existence of compact/complete metric on countable set
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
81927
Example of closed and bounded in $\mathbb Q$ set that is not compact
49374
Closure of a subset in a metric space
22131
Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
16826
prove that every infinite subset of compact set H has a limit point (Explanation)
125411
Infinite set always has a countably infinite subset
94485
Limit point compactness implies compactness.
66334
How is every subset of the set of reals with the finite complement topology compact?
88099
Not $\sigma$-compact set without axiom of choice
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
61930
Intersection of Closed and Compact Set is Closed
44974
Infinite set and countable subsets
67259
Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
29303
Cartesian product of compact sets is compact
144400
Are compact subsets of metric spaces closed and bounded?