Continuous map

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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:

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