Section 26: Problem 9 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
Suppose
and
are arbitrary spaces and
is compact, then if
open in the product contains
then it contains
for some neighborhood
of
and
of
. Indeed, each point
has a basis neighborhood
contained in
. The union of the basis neighborhoods covers the set which is homeomorphic to the compact set
and, therefore, there is a finite subcover. The finite intersection of
’s and the finite union of
’s are open and their product covers
and is contained in
.Now, for each
consider the basis element
that contains
and is contained within
. The union covers
and, since it is compact, there is a finite subcover. The finite union of
’s and the finite intersection of
’s are the open sets in
and
we are looking for.