Section 11*: Problem 5 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 Zorn’s lemma implies the following:
Lemma (Kuratowski). Let
be a collection of sets. Suppose that for every subcollection
of
that is simply ordered by proper inclusion, the union of the elements of
belongs to
. Then
has an element that is properly contained in no other element of
.
is strictly partially ordered by proper inclusion. Given a subcollection
of
, the union of the elements of
is an upper bound for
, which, by assumption, is also in
. Hence, Zorn’s lemma implies that there is a set in
such that no other set in
contains it.