If $(M,d)$ is a compact metric space and $f:M\rightarrow M$ is an isometry ($d(x,y)=d(f(x),f(y))$ for any $x,y\in M$), then $f$ is a homeomorphism
62479
A isometric map in metric space is surjective?
36566
Equivalence classes and metric spaces
82333
let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
20547
Homeomorphism that maps given points to given points
34699
A continuous bijection from a compact space to a $T_2$ space is always a homeomorphism
138607
Proof that a limit point compact metric space is compact.
90010
Can we ascertain that there exists an epimorphism $G\rightarrow H$?
128598
How to prove this is a metric?
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
14509
Contraction Map on Compact Normed Space has a Fixed Point
157597
Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$
39076
Contraction of compact sets
143043
When is a local homeomorphism a covering map?
66180
Homomorphism between compact Hausdorff spaces.
27517
Isometry in compact metric spaces
125982
If the dual spaces are isometrically isomorphic are the spaces isomorphic?
51227
Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$