Existence of maximizer implies compact?
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
10531
Is the convex hull of a compact set compact?
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
152493
Is an expanding map on a compact metric space continuous?
145415
Examples of almost compact spaces.
7081
isometry $f:X\to X$ is onto if $X$ is compact
116542
Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology]
144400
Are compact subsets of metric spaces closed and bounded?
19566
Continuous images of compact sets are compact
137203
Measurable cardinal existence condition
147331
Compactness and Hausdorffness
51914
Show that $S$ is compact.
93076
When $\min \max = \max \min$?
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
106957
Compact sets are closed?
22131
Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
5392
Compact support vs. vanishing at infinity?