Open Maps and Closed Maps Between Metric Spaces

# Open Maps and Closed Maps Between Metric Spaces

Definition: Let $X$ and $Y$ be metric spaces. A function $f : X \to Y$ is said to be an Open Map if for every open set $U \subseteq X$ we have that $f(U)$ is an open set in $Y$. Similarly, $f : X \to Y$ is said to be a Closed Map if for every closed set $C \subseteq X$ we have that $f(C)$ is a closed set in $Y$. |

*The terms "open" and "closed" used in the definitions above are relative to the spaces mentioned.*

Observe that if $f : X \to Y$ is a bijection then $f^{-1} : Y \to X$ exists. If $f$ is an open map then this implies that $f^{-1}$ is continuous by the open set criterion for continuity. Similarly, if $f$ is a closed map then this also implies that $f^{-1}$ is continuous by the closed set criterion for continuity.

Additionally, if $f : X \to Y$ is a bijection and if $f$ is continuous then $f^{-1}$ is an open map, and a closed map.