# Continuous map

## Contents

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

## Definition

### Symbol-free definition

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

### 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: