A continuous map between topological spaces f: X → Y is a function such that the preimage f⁻¹(V) is open in X whenever V is open in Y. Continuity means that f respects the topological structure: it pulls back observable properties in Y to observable properties in X.
This definition generalizes the ε-δ definition from analysis. In a metric space, f is continuous if and only if for every ε > 0 and every point x, there exists δ > 0 such that f maps the δ-ball around x into the ε-ball around f(x). The topological definition captures the same idea without requiring a metric: open sets encode “verifiability,” and continuity means verifiable properties of the output can be verified by examining the input.
Continuous maps are the morphisms of the category Top of topological spaces. Two spaces are equivalent in Top when there is a homeomorphism between them — a continuous bijection with continuous inverse. A continuous map that is not a homeomorphism may collapse or identify structure, like the map from a line segment to a circle.
In the categorical setting, a continuous map f: X → Y induces a geometric morphism between the sheaf topoi on X and Y: the direct image f_* pushes sheaves forward along f, and its left adjoint f* pulls them back. This is the bridge between point-set topology and topos theory — continuous maps between spaces become geometric morphisms between topoi.