Homotopy
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 {\displaystyle H:X\times \to Y} from the product of the space X with the unit interval to Y such that H (x, 0) = f (x) {\displaystyle H(x,0)=f(x)} and H (x, 1) = g (x) {\displaystyle H(x,1)=g(x)} for all x ∈ X {\displaystyle x\in 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. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa.

An alternative notation is to say that a homotopy between two continuous functions f, g: X → Y {\displaystyle f,g:X\to Y} is a family of continuous functions h t: X → Y {\displaystyle h_{t}:X\to Y} for t ∈ {\displaystyle t\in } such that h 0 = f {\displaystyle h_{0}=f} and h 1 = g {\displaystyle h_{1}=g}, and the map (x, t) ↦ h t (x) {\displaystyle (x,t)\mapsto h_{t}(x)} is continuous from X × {\displaystyle X\times } to Y {\displaystyle Y}. The two versions coincide by setting h t (x) = H (x, t) {\displaystyle h_{t}(x)=H(x,t)}. It is not sufficient to require each map h t (x) {\displaystyle h_{t}(x)} to be continuous.

The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R. X is the torus, Y is R, f is some continuous function from the torus to R that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; g is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of h (X) as a function of the parameter t, where t varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle.
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