Section 29: Problem 10 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
Let
where
is open and
is compact.
and
is open.
is closed in
which is compact (and closed), therefore,
is closed in
and compact. We can separate
and
by neighborhoods
and
. It follows that
is an open neighborhood of
in
such that
is a closed (and compact) subset of
.