Let $M$ be a metric space. If every continuous function $f : M \rightarrow \mathbb{R}$ has compact range then $M$ is compact.
9406
Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
36149
A uniformly continuous function maps bounded set to bounded sets
47810
$C(X)$ is separable when $X$ is compact?
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Is an expanding map on a compact metric space continuous?
30965
$f$ is continuous on $X$ iff $f$ is continuous on every compact subset
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isometry $f:X\to X$ is onto if $X$ is compact
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A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
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Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
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Continuous mapping on a compact metric space is uniformly continuous
55749
Compactly supported continuous function is uniformly continuous
147331
Compactness and Hausdorffness
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let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
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Which of the following sets are compact:
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Isometry in compact metric spaces
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Contraction of compact sets
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Proof that a limit point compact metric space is compact.
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If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.