Section 17: 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
Show that the product of two Hausdorff spaces is Hausdorff.
Take any two points
and
in the product space. If
, let
be any neighborhood of
, otherwise let
and
be disjoint neighborhoods of
and
. Similarly, if
, let
be any neighborhood of
, otherwise let
and
be disjoint neighborhoods of
and
. If
then
and
are disjoint and/or
and
are disjoint. Hence,
and
are disjoint neighborhoods of
and
, respectively.