Let $M$ be a compact metric space and $F:M\to M$ be a function that preserves distances, prove that is an homeomorphism
62479
A isometric map in metric space is surjective?
53812
Metric is continuous function
126759
Unbounded continuous function on non-compact metric space
90628
The metric spaces are Tychonoff spaces
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
74657
Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?
69777
Show that, for every metric $d$, the metrics $d/(1+d)$ and $\min\{1,d\}$ are equivalent
23845
How to prove the continuity of the metric function?
124179
Proof that every metric space is normal
7402
Showing a compact metric space has a countable dense subset
35573
A closed ball in a metric space is a closed set
46970
If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
154978
Some equivalent formulations of compactness of a metric space
18040
Show that a compact metric space is bounded
52014
Showing $(C[0,1], d_1)$ is not a complete metric space