Continuous map

This article is about a basic definition in topology.
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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:

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

Definition with symbols

Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ is termed continuous if $f$ satisfies the following equivalent conditions:

• $f^{-1}(U)$ is an open subset of $X$ for every open subset $U\subset Y$
• $f^{-1}(A)$ is a closed subset of $X$ for every closed subset $A\subset Y$

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