Supplementary Exercises*: Nets: Problem 6 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
If there is a net in
converging to
then for any neighborhood of
there is a point in
, therefore,
. Now, suppose
. Using 1(c), consider the collection
of all neighborhoods of
partially ordered by the reverse inclusion (the "finer" is the set, the "greater" it is). Now, for each neighborhood
take a point
. Then,
is a net of points of
converging to
. Indeed, given any neighborhood
of
and
for
:
.