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.