Definition:Homotopy

Let $$X$$ and $$Y$$ be topological spaces and $$f:X\to Y$$, $$g:X\to Y$$ be continuous maps. The two maps are said to be homotopic if there exists a continuous function $$H:X\times [0,1] \to Y$$ such that $$H(x,0)=f(x)$$ and $$H(x,1)=g(x)$$. $$H$$ is called a homotopy between $$f$$ and $$g$$.

A smooth homotopy is defined as above, with the word "continuous" replaced with "smooth."

Homotopy is an equivalence relation. The equivalence class of a function under homotopy is called its homotopy class.

= See also =

Homotopy is an Equivalence Relation