Section 19: Problem 3 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
Prove Theorem 19.4.
If two points
are different, then they have at least one different coordinate
. Then let
and
be disjoint open subsets of
containing
and
, respectively. We have,
and
, where
for
and
and
, are two disjoint open neighborhoods of
and
, respectively, in either topology.