Proof that if $(X,d)$ is a metric space and $\forall f:X\to \Bbb R, (f\text{ continuous} \implies f \text{ is bounded})$ then $X$ is compact
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
51914
Show that $S$ is compact.
113122
Prove that a continuous real function with finite limits is bounded
144372
How to Prove that if $A$ is bounded, then $A \cdot A$ is bounded?
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
14509
Contraction Map on Compact Normed Space has a Fixed Point
100121
Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.
99179
Continuous function with compact domain has continuous inverse
135157
Totally bounded, complete $\implies$ compact
143484
A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
53812
Metric is continuous function
118508
Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact?
153256
Compact topological space with closed graph implies continuity
152057
Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).
106534
"Closed subsets of compact space are compact" with an alternative definition of covering
147331
Compactness and Hausdorffness
154978
Some equivalent formulations of compactness of a metric space