Section 24: Problem 6 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
Indeed, for any two elements
there is an element between them (consider two cases
and
). We show the greatest lower bound property. If
is a nonempty set then
has the least element
and the greatest lower bound of
is
. By the way,
need not be a linear continuum (it is not iff
has an element that has a successor).