Homotopy (of maps)
A homotopy is a family of maps , , such that the associated map given by is continuous. Two maps are called homotopic, denoted , if there exists a homotopy connecting them.
Homotopy of paths
A homotopy of paths in a space is a family , , such that
(i) The endpoints and are independent of .
(ii) The associated map defined by is continuous.
When two paths and are connected in this way by a homotopy , they are said to be homotopic. The notation for this is .
The above two definitions are related, since a path is a special kind of map .