Retraction

A retraction $𝑟 : 𝑋 \to 𝑌$ is a continuous map from a topological space $𝑋$ to a subspace $𝑌 \subseteq 𝑋$ that does not move points in the subspace, i.e. $𝑟 (𝑦) = 𝑦$ for all $𝑦 \in 𝑌$ . Alternatively, if $𝜄 : 𝑌 ↪ 𝑋$ is the natural inclusion of the subspace topology, then a retraction $𝑟 : 𝑋 ↠ 𝑌$ is a continuous left-inverse of $𝜄$ , i.e. $𝑟 𝜄 = i d 𝑌$ . topology A subspace $𝑌 \subseteq 𝑋$ for which such a retraction exists is called a retract of $𝑋$ .

A special kind of retraction is a Deformation retraction, which has the additional property that $𝜄 𝑟 ≃ i d 𝑋$ .

Examples

The boundary of a ball is not a retract
Every one-point subset of $𝑋$ is a retract, but only a deformation retract if $𝑋$ is contractible.

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Retraction

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