Definition:Homotopy

Definition
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 Class
Homotopy is an Equivalence Relation. The equivalence class of a function under homotopy is called its homotopy class.