Compact subspace of $\mathbb{R}$ with lower limit topology must be countable.
106606
How to prove that compact subspaces of the Sorgenfrey line are countable?
44974
Infinite set and countable subsets
149317
An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
41047
Subspace Topology of a subset
70017
Every subspace of a compact space is compact?
116542
Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology]
94485
Limit point compactness implies compactness.
120135
Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets
149980
Are there any countable Hausdorff connected spaces?
22295
Subspace of a separable space is separable
105358
In a second countable space any base has a countable subset that is still a base
137067
For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.
153256
Compact topological space with closed graph implies continuity
6210
Countable compact spaces as ordinals
158498
$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
128730
Open subspaces of locally compact Hausdorff spaces are locally compact
22131
Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.