Supplementary Exercises*: Nets: Problem 11 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
Roughly speaking, if
then there is a net in
converging to
, there is a subnet of points in
converging to some
(
is compact and closed as
is Hausdorff), and the corresponding subnet of points in
converges to a point in
which must be
(
is closed, the operations are continuous).