# Open Mapping and Closed Mapping

Open Mapping

A mapping $f$ from one topological space $X$ into another topological space $Y$ is said to be an open mapping if for every open set $O$ in $X$, $f\left( O \right)$ is open in $Y$. In other words, a mapping is open if it carries open sets over to open sets. A continuous mapping may not be open.

Example:

Let $X = \left\{ {x,y,z} \right\}$ with topology ${\tau _X} = \left\{ {\phi ,X,\left\{ y \right\},\left\{ {x,y} \right\},\left\{ {y,z} \right\}} \right\}$ and let $Y = \left\{ {1,2,3} \right\}$ with topology ${\tau _Y} = \left\{ {\phi ,Y,\left\{ 1 \right\}} \right\}$. Then, $f:X \to Y$ is defined by $f\left( x \right) = 2$, $f\left( y \right) = 1$, $f\left( z \right) = 3$ is continuous but not open, because $A = \left\{ {x,y} \right\}$ is open in $X$ but $f\left( A \right) = \left\{ {1,2} \right\}$ is not open in $Y$.

Closed Mapping

A mapping $f$ from one topological space $X$ into another topological space $Y$ is said to be a closed mapping if for every closed set $G$ in $X$, $f\left( G \right)$ is closed in $Y$. In other words, a mapping is closed if it carries closed sets over to closed sets.

Theorems
• Let $X$ and $Y$ be topological spaces. A bijective mapping $f:X \to Y$ is open if and only if ${f^{ – 1}}:Y \to X$ is continuous.
• Let $X$ and $Y$ be topological spaces. A bijective mapping $f:X \to Y$ is closed if and only if ${f^{ – 1}}:Y \to X$ is continuous.
• Let $X$ and $Y$ be topological spaces. A mapping $f:X \to Y$ is open if and only if ${\left[ {f\left( A \right)} \right]^o} = f\left( A \right)$ for every open subset $A$ of $X$.