Notes:Algebraic Topology - Hatcher/Chapter 0

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Deformation retraction

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

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


  1. Here [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] of course