$C_{c}(X)$ is complete implies $X$ is compact.
64694
$C_{c}(X)$ is complete. then implies that $X$ is compact.
83311
Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
99645
Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
51853
$K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded.
8015
Compact set plus a closed set is a closed set
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
99927
The closure of $\{ z:|f(z)|<c\}$ is the set $\{z:|f(z)|\leq c\}$.
22131
Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
118765
A topological space is locally compact then here is an open base at each point has all of its set with compact closure
137067
For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.
88185
Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
146146
Proof that a closed subset of a compact space is a compact space
123061
$X$ metric separable then $C(X)$ separable
66334
How is every subset of the set of reals with the finite complement topology compact?
143674
Space of bounded functions is complete
148994
$K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
146455
Compactness of the set of all unitary matrices in $M_2(\mathbb{C})$