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{{basicdef}}

{{topospace map property}}

==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 <math>X</math> and <math>Y</math> be [[topological space]]s. A map <math>f:X \to Y</math> is termed '''continuous''' if <math>f</math> satisfies the following equivalent conditions:

* <math>f^{-1}(U)</math> is an open subset of <math>X</math> for every open subset <math>U \subset Y</math>
* <math>f^{-1}(A)</math> is a closed subset of <math>X</math> for every closed subset <math>A \subset Y</math>

==Facts==

===Category===

{{further|[[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]]

Continuous map - Revision history

Vipul: 1 revision

Vipul at 17:55, 11 December 2007