A homotopy between two continuous maps 
is a continuous map ![G:X×[0,1]->Y](/images/equations/Homotopy/Inline2.svg)
such that 
and 
, where 
and 
are topological spaces and 
denotes the Cartesian
product. A homotopy also defines a path from 
to 
in the function space 
with the compact-open
topology.
Two continuous maps with the same domain and codomain are homotopic iff a homotopy connects them. The corresponding notion for topological spaces is homotopy equivalence. Around 1900, Poincaré used continuous deformations of loops to study topological properties (Collins 2004, p. 99).
