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In Algebraic topology by Hatcher, there is a statement saying that
A homotopy $f_t:X\rightarrow X$ that gives a deformation retraction of $X$ onto a subspace $A$ has the property that $f_t(A)=identity$ for all $t$.
I don't understand why this happens. In some steps of deformation retraction the functions $f_t$ many not be identical on $A$, we only look in the initial and final steps. I have a big confusion about it.
Please help me to understand that line.

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  • $\begingroup$ See his definition of deformation retraction in Chapter 0. $\endgroup$ – Lord Shark the Unknown Mar 3 at 6:21
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This terminology varies some across authors. Often an author will define a deformation retraction of $X$ onto $A \subset X$ to be a continuous map $X \times [0, 1] \to X$ satisfying (for all $x \in X$, $a \in A$) $$H(x, 0) = x, \qquad H(x, 1) \in A, \qquad H(a, 1) = a$$ and define a strong deformation retraction to be a deformation retraction that additionally satisfies $$H(a, t) = a$$ for all $a \in A$, $t \in [0, 1]$ (obviously this condition implies the condition that $H(a, 1) = a$ for all $a \in A$). But some authors, including Hatcher (see $\S$ 0, page 2), define a deformation retraction to satisfy this last condition, too.

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  • $\begingroup$ Thank you for your answer. My confusion is clear now. $\endgroup$ – chandan mondal Mar 3 at 6:37
  • $\begingroup$ You're welcome, I'm glad I could clarify for you. $\endgroup$ – Travis Mar 3 at 7:06

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