Deformation retraction/Definition

Definition

A subspace, $A$, of a topological space $(X,\mathcal{ J })$ is called a deformation retract of $X$, if there exists a retraction, $r:X\rightarrow A$, with the additional property:

• $i_A\circ r\simeq\text{Id}_X$ (That $i_A\circ r$ and $\text{Id}_X$ are homotopic maps)

  Here $i_A:A\hookrightarrow X$ is the inclusion map and $\text{Id}_X$ the identity map of $X$.

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

Caution:Be sure to see the warnings on terminology

References

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TODO: Mention something about how we must have a homotopy equivalence as a result. If $r\circ i_A=\text{Id}_A$ then $r\circ i_A$ and $\text{Id}_X$ are trivially homotopic. As $i_A\circ r\simeq\text{Id}_A$ we have the definition of a homotopy equivalence

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