Existence of Isometric Spaces

tags:theorem |topology

Existence of Isometric Spaces

Statements

Definition (@Men90, Theorem 7.4). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, if there exists a function $f:X\to Y$ such that

1. $f$ is injective.
2. $f$ is surjective.
3. For every $x_1,x_2\in X$, $d_Y(f(x_1),f(x_2))=d_X(x_1,x_2)$. Then, $(X,d_X)$ and $(Y,d_Y)$ are said to be isometric spaces.