Difference between revisions of "Notes:Algebraic Topology - Hatcher/Chapter 0"

Latest revision as of 20:57, 1 November 2016

Book notes

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.

Notes

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