tags:definition |topology

Isometric Spaces

Statements

Definition (@Men90 Definition 7.3). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, if there exist inverse functions $f:X\to Y$ and $g:Y\to X$ such that, for every $x_1,x_2\in X$, $d_Y(f(x_1),f(x_2))=d_X(x_1,x_2)$, and for every $y_1,y_2\in Y$, $d_X(g(y_1),g(y_2))=d_Y(y_1,y_2)$. Then, $(X,d_x)$ and $(Y,d_Y)$ are said to be metrically equivalent or isometrics by $f$ and $g$.