Continuous map of topological spaces

Definition

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces and let $f \colon X \to Y$ be a function. The function $f$ is continuous when for each open set $V \in \tau_Y$, the preimage $f^{-1}(V)$ is in $\tau_X$.

Remarks

Equivalently, $f$ is continuous if and only if preimages of closed sets are closed, as in continuity via closed sets. For metric spaces, this is the same as the epsilon-delta condition, by continuity via open sets. A function is continuous if and only if it is continuous at each point. Constant maps and identity maps are continuous, and the composition of continuous maps is continuous.

Explanations

L1 human

A continuous map does not tear the space apart: the points that go into an open set come from an open set. This definition uses only open sets, so it works in each topological space.