TOPICS
Search

Deformation Retract


A subspace A of X is called a deformation retract of X if there is a homotopy F:X×I->X (called a retract) such that for all x in X and a in A,

1. F(x,0)=x,

2. F(x,1) in A, and

3. F(a,1)=a.

A tightening of the last condition gives a so-called strong deformation retract (Bredon 1993, pp. 45-46).

Note that a deformation retract is also a retract, because the homotopy F:X×I->X defines a continuous map f:X->X

 f(x)=F(x,1).

See also

Deformation, Retract, Strong Deformation Retract

Portions of this entry contributed by John Renze

Portions of this entry contributed by Jonathan Sondow (author's link)

Explore with Wolfram|Alpha

References

Bredon, G. E. Topology and Geometry. New York: Springer-Verlag, 1993.Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2001.

Referenced on Wolfram|Alpha

Deformation Retract

Cite this as:

Renze, John; Sondow, Jonathan; and Weisstein, Eric W. "Deformation Retract." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeformationRetract.html

Subject classifications