Exercise 58.1

Question

Show that if $A$ is a deformation retract of $X$, and $B$ is a deformation retract of $A$, then $B$ is a deformation retract of $X$.

Amit Awasthi · Accepted Answer

\begin{proof}
  Let $F\colon X 	imes I 	o X$, $G\colon A 	imes I 	o A$ be the deformation retractions of $X$ onto $A$ and $A$ onto $B$ respectively. We claim that
  \begin{equation*}
    H(x,t) = \begin{cases}
      F(x,2t) & 	ext{if}~0 \le t \le 1/2\
      \iota \circ G(F(x,1),2t-1) & 	ext{if}~1/2 \le t \le 1
    \end{cases}
  \end{equation*}
  is a deformation retraction of $X$ onto $B$, where $\iota\colon A \hookrightarrow X$ is the inclusion map. We see $H(x,0) = F(x,0) = x$, and that if $x \in B$, $H(x,1) = \iota \circ G(F(x,1),1) = \iota \circ F(x,1) = \iota \circ x = x$; moreover, $H$ is continuous by the pasting lemma, since $H(x,1/2) = F(x,1) = \iota \circ G(F(x,1),0) = \iota \circ F(x,1) = F(x,1)$, and since $\iota \circ G(F(x,1),2t-1)$ is a composition of continuous functions.
\end{proof}

Exercise 58.1

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Exercise 58.1

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