# nLab closed map

### Context

#### Topology

topology

algebraic topology

# Contents

## Definition

### For maps between topological spaces

A function $f:X\to Y$ between topological spaces is called closed if the image of every closed set in $X$ is also closed in $Y$.

Recall that $f$ is a continuous map if the preimage of every closed set in $Y$ is closed in $X$. For defining closed maps typically one restricts attention to closed continuous maps, although it also makes sense to speak of closed functions that are not continuous.