Strong deformation 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 deformation retract if there is a homotopy between the identity map on the whole space, and a retraction onto the subspace, such that the map at every intermediate stage, restricts to identity on the subspace.

Definition with symbols

A subspace $A$ of a topolofical space $X$ is termed a deformation retract of $X$ if there is a homotopy $F:X\times I\to X$ such that:

• $f(x,0)=x\mathrm{\forall }x\in X$
• $f(a,t)=a\mathrm{\forall }a\in A,t\in I$
• $f(x,1)\in A\mathrm{\forall }x\in X$

The second condition is what distinguishes deformation retracts from the weaker notion of homotopy retract.

Relation with other properties

Weaker properties

• Homotopy retract
• Retract