Deformation retract

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"Retract" redirects here. For other meanings including concepts in group theory and category theory, see Retraction (disambiguation).





r:X \to A


\iota : A \hookrightarrow X

the inclusion, a retraction is a continuous map r such that

r \circ \iota = id_A,


If r:X \to A is a retraction, then the composition \iota \circ r is an idempotent continuous map from X to X. Conversely, given any idempotent continuous map s:X\to X, we obtain a retraction onto the image of s by restricting the codomain.


Neighborhood retract[edit]

If there exists an open set U such that

A \subset U \subset X



Deformation retract and strong deformation retract[edit]

A continuous map

F:X \times [0, 1] \to X \,


 F(x,0) = x, \; F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a.





F(a,t) = a\,



F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.

Neighborhood deformation retract[edit]

A closed subspace A is a neighborhood deformation retract of X if there exists a continuous map u:X \rightarrow I (where I=[0,1]) such that A = u^{-1} (0) and a homotopy H:X\times I\rightarrow X such that H(x,0) = x for all x \in X, H(a,t) = a for all (a,t) \in A\times I, and h(x,1) \in A for all x \in u^{-1} [ 0 , 1).[2]


  • The main obvious property of a retract A of X is that any continuous map f : A \rightarrow Y has at least one extension g : X \rightarrow Y, namely, g=f\circ r\,.


  1. ^ K. Borsuk (1931). "Sur les rétractes". Fund. Math. 17: 2–20. 
  2. ^ Steenrod, N. E. (1967). "A convenient category of topological spaces". Michigan Math. J. 14 (2): 133–152. doi:10.1307/mmj/1028999711. 
  3. ^ Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1 


External links[edit]