For $(X,d)$ a compact metric space and $ Y \subsetneq X$ there is no isometry $f:X \to Y$.
74657
Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
55016
Prove that if ($d(x,y)=0$ iff $x=y$) and if $d(x,z)\leq d(x,y)+d(z,y)$ then $d$ is a metric
100121
Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.
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).
29303
Cartesian product of compact sets is compact
64261
example of not surjective isometry on a bounded not closed set.
37654
Topology: Example of a compact set but its closure not compact
26794
Metrizability of Hausdorff continuous image of compact metric space
151984
Proving the existence of a metric space
35652
Every compact metric space is complete
110485
Nested sequence of non-empty compact subsets - intersection differs from empty set
20706
Non-existence of a Surjective Function from a Set to Its Subsets (Cantor's theorem)
32819
Looking for examples of first countable, compact spaces which is not separable
39076
Contraction of compact sets
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
123061
$X$ metric separable then $C(X)$ separable