Pseudocompactness implies Compactness in metric spaces
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
28637
Proving continuous image of compact sets are compact
102048
Closure of a compact set in Regular space $ X $
7402
Showing a compact metric space has a countable dense subset
112303
Distances between closed sets on metric spaces
153256
Compact topological space with closed graph implies continuity
94485
Limit point compactness implies compactness.
94438
Show that the countable product of metric spaces is metrizable
9406
Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
128897
Bounded sequence in a metric space
84004
Prove there is no contraction mapping from compact metric space onto itself
145415
Examples of almost compact spaces.
28804
No. of possible dense subsets of a metric space
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
155621
Finiteness of the dimension of a normed space and compactness
29303
Cartesian product of compact sets is compact
71916
Quasi-components and components coincide for compact Hausdorff spaces