Section 54: Problem 2 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
In defining the map
in the proof of Lemma 54.2, why were we so careful about the order in which we considered the small rectangles?
I don’t think the order itself matters, however, whatever the order is chosen it must be such that when the lifting of the path is about to be constructed on a given rectangle, it must already be constructed on some non-empty and connected subset of the rectangle; this ensures both, that the path can be constructed, and can be constructed in a unique way.