Section 11*: Problem 3 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 set with a strict partial order
; let
. Suppose that we wish to find a maximal simply ordered subset
of
that contains
. One plausible way of attempting to define
is to let
equal the set of all those elements of
that are comparable with
;
But this will not always work. In which of Examples 1 and 2 will this procedure succeed and in which will it not?
On the one hand, this will definitely work if the set can be partitioned into a collection of disjoint ordered subsets such that no two elements from two different subsets are comparable. On the other hand, this will fail if, for example,
is comparable with two elements that are not comparable. So, for the set in Example 1 this will fail (every circle is comparable with two larger circles that are not comparable), and for the sets in Example 2 and Exercise 1 it will succeed.