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:
“That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.”
—Fred Rogers (20th century)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)