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
Famous quotes containing the words formal and/or definition:
“Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more formal one said, “Do you think the Holy Dove could fly down with only one wing?””
—Horace Walpole (1717–1797)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)