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Deformation retract

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Let $X$ be a topological space and $A$ a subspace of $X$. A continuous function $r:X \to A$ such that $r(x)=x$ for all $y \in A$ is called a retraction. We also say that $A$ is a retract of $X$. Map $r$ is a deformation retraction of $X$ if there is a collection of maps $f_t:X\rightarrow X$, $t\in [0,1]$, such that

  • $f_0 = Id_X$, the identity map on $X$,
  • $f_1$ is a retraction of $X$ to $A$,
  • the function $F:X\times I\rightarrow X$, given by $F(x,t)=f_t(x)$ is continuous, where $X\times I$ has the product topology.

Map $F$ is a homotopy rel A between $Id_X$ and $r$.

Exercises.

  • A convex set deformation retracts to any of its points.
  • ${\bf R}^n\backslash \{0\}$ deformation retracts to the $(n-1)$-sphere ${\bf S}^{n-1}$.
  • The Mobius band $M^2$ deformation retracts to the circle ${\bf S}^1$.
  • The torus $T^2$ with point removed deformation retracts to the union of the equator and a meridian (bouquet of two circles ${\bf S}^1$).