# Deformation retraction

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## Definition

A subspace, [ilmath]A[/ilmath], of a topological space [ilmath](X,\mathcal{ J })[/ilmath] is called a *deformation retract* of [ilmath]X[/ilmath], if there exists a retraction^{[1]}^{[2]}, [ilmath]r:X\rightarrow A[/ilmath], with the additional property:

- [ilmath]i_A\circ r\simeq\text{Id}_X[/ilmath]
^{[1]}^{[2]}(That [ilmath]i_A\circ r[/ilmath] and [ilmath]\text{Id}_X[/ilmath] are homotopic maps)- Here [ilmath]i_A:A\hookrightarrow X[/ilmath] is the inclusion map and [ilmath]\text{Id}_X[/ilmath] the identity map of [ilmath]X[/ilmath].

Recall that a retraction, [ilmath]r:X\rightarrow A[/ilmath] is simply a continuous map where [ilmath]r\vert_A=\text{Id}_A[/ilmath] (the restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath]). This is equivalent to the requirement: [ilmath]r\circ i_A=\text{Id}_A[/ilmath].

## Warnings on terminology

Some authors define a *deformation retract* to be what we would call a strong deformation retraction.

TODO: Make a table or something

## References

- ↑
^{1.0}^{1.1}An Introduction to Algebraic Topology - Joseph J. Rotman - ↑
^{2.0}^{2.1}Introduction to Topological Manifolds - John M. Lee