##  [Deformation Retraction](/deformation-retraction) 

  ##  [Deformation Retraction](https://mathlogic.quantumdictionary.io/deformation-retraction-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A homotopy H: X × [0,1] → X that deforms a topological space X onto a subspace A so that H(x,0)=x for all x, H(x,1)∈A for all x, and H(a,t)=a for all a∈A and t∈[0,1]; in other words, a continuous deformation of X that leaves A fixed and ends with a retraction onto A.

 

 

 

 

 





 

 



 ##  [Deformation Retraction](https://puremath.quantumdictionary.io/deformation-retraction-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A homotopy H: X × [0,1] → X that continuously deforms a topological space X onto a subspace A so that H(x,0)=x, H(x,1) ∈ A for all x in X, and H(a,t)=a for all a in A and all t; such a homotopy exhibits A as a deformation retract of X and gives a strong homotopy equivalence.