Types of topological retractions

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Definitions

Retraction

Let $(X,\mathcal{ J })$ be a topological space and let $A\in\mathcal{P}(X)$ be considered a s subspace of $X$. A continuous map, $r:X\rightarrow A$ is called a retraction if:

• The restriction of $r$ to $A$ (the map $r\vert_A:A\rightarrow A$ given by $r\vert_A:a\mapsto r(a)$) is the identity map, $\text{Id}_A:A\rightarrow A$ given by $\text{Id}_A:a\mapsto a$

If there is such a retraction, we say that: $A$ is a retract of $X$.

Deformation retraction

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

Strong deformation retraction

Strong deformation retraction/Definition