Let be a topological space and be homeomorphisms from to itself. An isotopy from to is a homotopy that starts at , ends at , and such that for any fixed , the map is a homeomorphism.
Loosely, it is a homotopy via homeomorphisms.
A composite of isotopies gives an isotopy, and hence being isotopic defines an equivalence relation on the group of all homeomorphisms. The quotient of the group of all homeoomorphisms by this equivalence relation gives the mapping class group of the topological space.