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


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 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



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