12. Isometries and Homeomorphisms

Isometry between Metric Spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces then
Function $f:X\to Y$ between $(X,d_X)$ and $(Y,d_Y)$ is an isometry if

$$d_Y(f(x),f(y))=f_X(x,y)\text{ for all }x,y\in X$$

Homeomorphisms

Let $f:X\to Y$ be a continuous function between metric space $X$ and $Y$

$f$ is a homeomorphism if

1. Continuous
2. A bijection
3. Inverse $f^{-1}:Y\to X$ is continuous