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 strong deformation retract (sometimes simply 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. Such a homotopy is termed a strong deformation retraction.

Definition with symbols

A subspace $A$ of a topological space $X$ is termed a strong deformation retract (sometimes simply deformation retract) of $X$ if there is a homotopy $F:X\times I\to X$ such that:

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

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

Relation with other properties

Weaker properties

• Homotopy retract
• Retract
• Neighbourhood retract

Metaproperties

Transitivity

This property of subspaces of topological spaces is transitive. In other words, if $A$ satisfies the property as a subspace of $B$ and $B$ satisfies the property as a subspace of $C$ then $A$ satisfies the property as a subspace of $C$

If $A$ is a strong deformation retract of $B$ and $B$ is a strong deformation retract of $C$ then $A$ is a deformation retract of $C$.

If $A_{i}$ is a deformation retract of $B_{i}$ for $i=1,2$ then $A_{1}\times A_{2}$ is a deformation retract of $B_{1}\times B_{2}$.