Notes:Algebraic Topology - Hatcher/Chapter 0

Book notes

Deformation retraction

A deformation retraction of a topological space $(X,\mathcal{ J })$ onto a subspace $(A,\mathcal{J}_A)$ is a family of maps:

• For all $t\in I$:
  • $f_t:X\rightarrow X$ such that
    1. $f_0=\text{Id}_X$, the identity map,
    2. $f_1(X)=A$ and
    3. $f_t\big\vert_A=\text{Id}_A$
• The family $\{f_t\}_{t\in I}$ should be continuous in the sense that the associated map:
  • $(:X\times I\rightarrow X)$ given by $(:(x,t)\mapsto f_t(x))$ is continuous.