# Continuous map

This article is about a basic definition in topology.VIEW: Definitions built on this | Facts about this | Survey articles about this

View a complete list of basic definitions in topology

*This article defines a property that can be evaluated for a map between topological spaces. Note that the map is not assumed to be continuous*

## Contents

## Definition

### Symbol-free definition

A map from one topological space to another is termed **continuous** if it satisfies the following equivalent conditions:

- The inverse image of every open subset is open
- The inverse image of every closed subset is closed

### Definition with symbols

Let and be topological spaces. A map is termed **continuous** if satisfies the following equivalent conditions:

- is an open subset of for every open subset
- is a closed subset of for every closed subset

## Facts

### Category

`Further information: Category of topological spaces with continuous maps`

Continuous maps are the *morphisms* in the category of topological spaces. In particular, the identity map is continuous, and a composite of continuous maps is also continuous.

For a list of properties that continuous maps may or may not satisfy, refer:

Category:Properties of continuous maps between topological spaces