# Open Mapping and Closed Mapping

**Open Mapping**

A mapping from one topological space into another topological space is said to be an open mapping if for every open set in , is open in . 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 with topology and let with topology . Then, is defined by , , is continuous but not open, because is open in but is not open in .

**Closed Mapping**

A mapping from one topological space into another topological space is said to be a closed mapping if for every closed set in , is closed in . In other words, a mapping is closed if it carries closed sets over to closed sets.

**Theorems**

• Let and be topological spaces. A bijective mapping is open if and only if is continuous.

• Let and be topological spaces. A bijective mapping is closed if and only if is continuous.

• Let and be topological spaces. A mapping is open if and only if for every open subset of .