Let be a manifold and the closed unit interval. A smooth map is called a diffeotopy (on ) if for every :

Two diffeomorphisms are said to be diffeotopic if there is a diffeotopy such that

1. , and
2. .

Remark. Diffeotopy is an equivalence relation among diffeomorphisms. In particular, those diffeomorphisms that are diffeotopic to the identity map form a group.

Two points are said to be isotopic if there is a diffeotopy on such that

1. , the identity map on , and
2. .

Remark. If is a connected manifold, then every pair of points on are isotopic.

Pairs of isotopic points in a manifold can be generazlied to pairs of isotopic sets. Two arbitrary sets are said to be isotopic if there is a diffeotopy on such that

1. , and
2. .

Remark. One special example of isotopic sets is the isotopy of curves. In , curves that are isotopic to the unit circle are the trivial knots.