In mathematics, a pointed space is a topological space X with a distinguished basepoint x₀ in X. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f(x₀) = y₀. This is usually denoted

f : (X, x₀) → (Y, y₀).

Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint.