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So I'm trying to understand the proof on page 63

http://www.math.cornell.edu/~hatcher/AT/AT.pdf

In the proof he says that if $\tilde{f}_{1}(y) \not = \tilde{f}_2 (y)$, then $\tilde{U}_{1} \not = \tilde{U}_2$. But,how is this true? You can't deduce this from anything. Certainly, there must be something magical happening for him to deduce this. Surely, they the maps could be taken the points to different places in $\tilde{U}_1, \tilde{U}_2$, but they are still equal.

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    It tends to be nicer to phrase this type of questions more along the line of «I do not understand such line in Hatcher» than «Hatcher seems to be wrong»... :)2012-05-21
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    I would not normally upvote this question, but while the choice of words is somewhat unfortunate, I can’t see downvoting the question on that account; accordingly, I’ve upvoted.2012-05-21
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    Prof Hatcher [still maintains](http://www.math.cornell.edu/~hatcher/AT/corrections.html) the book, which seems to be used by everyone and their dog; so it sounds unlikely that there's hocus pocus in the first chapter!2012-05-21

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