Homotopy - Formal Definition

Formal Definition

Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X × → Y from the product of the space X with the unit interval to Y such that, if x ∈ X then H(x,0) = f(x) and H(x,1) = g(x).

If we think of the second parameter of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g.

An alternative notation is to say that a homotopy between two continuous functions f, g : X → Y is a family of continuous functions h_t : X → Y for t ∈ such that h₀ = f and h₁ = g, and the map t ↦ h_t is continuous from to the space of all continuous functions X → Y. The two versions coincide by setting h_t(x) = H(x,t).