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Let $p:E \rightarrow B$ a fibration and take $A\subset B$ a deformation retract of B. Is it true that $p^{-1}(A)$ is a deformation retract of E?

By deformation retract I mean the weaker definition.

I've seen this answer before as a direct use of the definition of a fibration, but I don't see it that clear.

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    $\begingroup$ Hello, and welcome to MathOverflow! Your question was asked here several years ago, and has a great answer: mathoverflow.net/questions/178509/…. It's a great question, and I hope the answer there helps you. Because we prefer not to have duplicate questions, I'll vote to close this one as a duplicate of the other. $\endgroup$
    – David White
    Commented Mar 13 at 20:07
  • $\begingroup$ I know, I've read this answer a lot of times, but I don't really understand the answer at all $\endgroup$
    – Alvaro Sopeña
    Commented Mar 13 at 21:29
  • $\begingroup$ Please read Allen Hatcher's answer. That one focuses on deformation retractions. The other answers are about strong deformation retractions. The answer to your question is "yes" and Hatcher explains why. Just draw the lifting diagram he describes. $\endgroup$
    – David White
    Commented Mar 13 at 22:29

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