Continuous map - Imported Knowlpedia Content

A continuous map between topological spaces $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ is a function $f:X\to Y$ such that for every open set $V\in\mathcal{T}_Y$ , the preimage $f^{-1}(V)$ is open in $X$ .

Equivalent characterizations

Equivalently, $f$ is continuous if the preimage of every closed set in $Y$ is closed in $X$ .

Remarks

In practice, continuity is often checked using a basis or subbasis of the topology on $Y$ .

Examples

The identity map $\mathrm{id}_X:X\to X$ is continuous for any topological space $X$ .
Any constant map $f:X\to Y$ (sending all of $X$ to a single point of $Y$ ) is continuous.
If $Y\subseteq X$ has the subspace topology, the inclusion map $i:Y\to X$ is continuous.