Section 17: Problem 1 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
be a collection of subsets of the set
. Suppose that
and
are in
, and that finite unions and arbitrary intersections of elements of
are in
. Show that the collection
is a topology on
.
The proof is similar to Theorem 17.1, just the other direction. The empty set and
are in
because they are complements of
and
, respectively, which are in
. And the complement
, where
is an arbitrary union (a finite intersection) of some elements of
, is the intersection (the finite union) of the complements of these elements, which belong to
, therefore,
.