In Hatcher's Algebraic Topology p.30, in the last but one paragraph, he said: After replacing N by a smaller neighborhood of $y_0$, we may assume that $\tilde{F}(N\times t_i)$ is contained in $\tilde{U_i}$, namely replace $N\times t_i$ by its intersection with ${\tilde{F}|(N\times t_i)}^{-1}$($\tilde{U_i}$).

But I wonder if the replacement is necessary. The reason is: By a former claim in the same paragraph:$F(N\times t_i)\subset U_i$, and $F(N\times t_i)\subset U_{i-1}$. So to the $\tilde{U_i}$ contains $y_0\times t_i$, $\tilde{F}(N\times t_i)$ must be in $\tilde{U_i}$. i.e. the intersection of intersection with ${\tilde{F}|(N\times t_i)}^{-1}$($\tilde{U_i}$) and $N\times t_i$ is $N\times t_i$. Do I make some mistakes above? Thank you!

  • $\begingroup$ Probably you should explain the proof you are studying more in depth, with some definitions etc. Just my opinion. $\endgroup$ – Maffred Jan 29 '16 at 0:37
  • $\begingroup$ You could, for example, google for the pdf of the book and copy/paste the part you need on paint then upload here. $\endgroup$ – Maffred Jan 29 '16 at 0:43
  • $\begingroup$ It seems that is Hathcer himself who answered the question! ^^ $\endgroup$ – Maffred Jan 29 '16 at 0:47
  • $\begingroup$ @Maffred thank you so much! $\endgroup$ – 6666 Jan 29 '16 at 1:00