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 ht : X → Y for t ∈ such that h0 = f and h1 = g, and the map t ↦ ht is continuous from to the space of all continuous functions X → Y. The two versions coincide by setting ht(x) = H(x,t).
Read more about this topic: Homotopy
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