Continuous map

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 $$A \subset Y$$

Category
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