Deformation retraction

This article defines a property of a homotopy from a topological space to itself

Definition

Let $X$ be a topological space and $A$ a subspace. A deformation retraction from $X$ to $A$ is a homomotopy $F:X\times I\to X$ such that:

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

If a deformation retraction exists from $X$ to $A$, we say that $A$ is a deformation retract of $X$.

Relation with other properties

Stronger properties

• Sudden deformation retraction
• Semi-sudden deformation retraction

Weaker properties

• Weak deformation retraction