# 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 $$X$$.

Yatin Kumar

May 18@ 5:43 pmProofs of these theorems