Section 26: Problem 5 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
We know from Lemma 26.4 that for each
there are open sets
and
such that
.
covers
and there is a finite subcovering
. The finite intersection of corresponding sets
is open, contains
and does not intersect
. (The proof is also given as a part of the proof of Theorem 32.3.)