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
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
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: