Pointed space

In mathematics, a pointed space is a topological space with a distinguished point, the basepoint. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f between a pointed space X with basepoint x₀ and a pointed space Y with basepoint y₀ is a based map if it is continuous with respect to the topologies of X and Y and if f(x₀) = y₀. This is usually denoted