Homotopy

A continuous transformation from one function to another. A homotopy between two functions  and  from a space  to a space  is a continuous map  from  such that  and , where  denotes set pairing. Another way of saying this is that a homotopy is a path in the mapping space  from the first function to the second.

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. The concept of homotopy was first formulated by Poincaré around 1900 (Collins 2004).

REFERENCES:

Aubry, M. Homotopy Theory and Models. Boston, MA: Birkhäuser, 1995.

Collins, G. P. "The Shapes of Space." Sci. Amer. 291, 94-103, July 2004.

Krantz, S. G. "The Concept of Homotopy" §10.3.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 132-133, 1999.