I have a question about Hatcher's proof that the fundamental group of a circle is Z. Specifically, halfway through, ( http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf , page 30), he proves an important a lemma stated rather generally:
Given a map $Y\times I\rightarrow S^1$ (Y is any space, and I is an interval), there is a unique lifting to $Y\times I\rightarrow \mathbb{R}$ once we've specified an initial condition $Y\times \{0\}\rightarrow \mathbb{R}$.
In the proof of this, he breaks $Y\times I$ into smaller pieces $N\times \{y_0\}$, and keeps adjusting the size of the $N$. In the corresponding proofs in Munkres or Fulton, $Y$ is fixed as just another copy of $I$, and there is no need to modify the pieces we're looking at as the proof progresses.
My question: what is different in Hatcher's more general proof? Specifically, why does he keep modifying the $N$? My feeling is that this is because $Y$ could be disconnected or bad in some other sense. My question seems a little vague, but (a) if you look at Hatcher's proof, you'll see what I mean (near the bottom of p. 30), and (b) I'm hoping that maybe this situation, and the difficulties with it, are well known enough for someone to answer anyway.