Section 11*: Problem 7 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 Tukey lemma implies the Hausdorff maximum principle. [Hint: If
is a strict partial order on
, let
be the collection of all subsets of
that are simply ordered by
. Show that
is of finite type.]
Following the hint, let
be the collection of ordered subsets of
. If a set
is in
, it is ordered, and all its finite subsets are ordered as well, so all finite subsets of
are in
. Now, suppose that for a set
all its finite subsets are in
. Then for any two different elements of
, the set consisting of these two elements is finite, hence, is in
and ordered. Therefore, the set
is ordered as well. It follows that
is of finite type. Therefore, Tukey’s lemma implies that there is a maximal ordered set in the collection, as no other ordered set (no other set in
) contains it.