A homotopy class is an equivalence class of continuous maps 
between two topological
spaces 
and 
under homotopy. Write 
if 
is homotopic to 
. Roughly speaking, two maps are
homotopic if one can be deformed into the other. This
equivalence relation is transitive
because these homotopy deformations can be composed
(i.e., one can follow the other).
A simple example is the case of continuous maps from one circle to another circle.
Consider the number of ways an infinitely stretchable string can be tied around a
tree trunk. The string forms the first circle, and the
tree trunk's surface forms the second circle. For any
integer 
, the string can be wrapped around the tree 
times, for positive 
clockwise, and negative 
counterclockwise. Each integer 
corresponds to a homotopy class of maps from 
to 
.
After the string is wrapped around the tree 
times, it could be deformed a little bit to get another continuous map, but it would still be in the same
homotopy class, since it is homotopic to the original
map. Conversely, any map wrapped
around 
times can be deformed to any other.
