# Notes:Algebraic Topology - Hatcher/Chapter 0

From Maths

## 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- [ilmath]f_0=\text{Id}_X[/ilmath], the identity map,
- [ilmath]f_1(X)=A[/ilmath] and
- [ilmath]f_t\big\vert_A=\text{Id}_A[/ilmath]

- [ilmath]f_t:X\rightarrow X[/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

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