Closed Map

Definition

A function $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between two topological spaces is a closed map if the function value of a closed set is a closed set. $(X\setminus C)\in {\mathcal{T}}_X⟹(Y\setminus f(U))\in {\mathcal{T}}_Y$

Facts

Consider a Continuous Function $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between two topological spaces. If $X$ is aCompact space, and $Y$ is a Hausdorff Space $(Y,{\mathcal{T}}_Y)$, then $f$ is a closed map.