Precompact and bounded, Metric Space
45405
An elementary way to show any bounded subset of $\Bbb{R}^k$ is totally bounded
95783
Fixed points of contractions in metric spaces
138607
Proof that a limit point compact metric space is compact.
29744
Compactness and axiomatisability
70106
diameter on a compact metric space
49563
Cauchy nets in a metric space
144400
Are compact subsets of metric spaces closed and bounded?
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
126759
Unbounded continuous function on non-compact metric space
80276
Metric Space & Cauchy Sequence
126944
Metric spaces problem
99645
Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
64260
Prove that a function between metric spaces is continuous iff the inverse image of a closed set is closed
35573
A closed ball in a metric space is a closed set
119105
If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
151832
uniformly convergence on compact metric space
153952
A map between metric spaces preserving convergent sequences is continuous