deformation retract

Definition

A deformation retract is a retract such that there is a homotopy from the identity map to the retract.

In other words, a deformation retract is a continuous map $r:X\to A$ from a topological space $X$ to a subspace $A$ such that

$r{|}_A=i{d}_A$ and there exists a homotopy

$H:B\times I⟶B$

such that

$H(a,t)=a\forall a\in A,t\in T,\text{and}H(x,1)\in A\forall x\in X.$