Homotopy retract

This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces

Definition

Symbol-free definition

A subspace of a topological space is termed a homotopy retract if the identity map from the whole space to itself is homotopic to the retraction onto that subspace.

Definition with symbols

A subspace $A$ of a topological space $z$ is termed a homotopy retract of $X$ if there exists a map $F:X\times I\to X$ such that:

• $F(x,0)=x\mathrm{\forall }x\in X$
• $F(x,1)\in A\mathrm{\forall }x\in X$

Note that unlike in the stronger notion of deformation retract, we do not require that at intermediate times, $F$ should restrict to the identity on $A$.

Relation with other properties

Stronger properties

• Deformation retract

Weaker properties

• Retract