Deformation Retraction

A homotopy H: X × [0,1] → X that deforms a topological space X onto a subspace A so that H(x,0)=x for all x, H(x,1)∈A for all x, and H(a,t)=a for all a∈A and t∈[0,1]; in other words, a continuous deformation of X that leaves A fixed and ends with a retraction onto A.

A homotopy H: X × [0,1] → X that continuously deforms a topological space X onto a subspace A so that H(x,0)=x, H(x,1) ∈ A for all x in X, and H(a,t)=a for all a in A and all t; such a homotopy exhibits A as a deformation retract of X and gives a strong homotopy equivalence.