Continuous map

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 $X$.

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