Supplementary Exercises*: Well-Ordering: Problem 2 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
(a) Let
and
be well-ordered sets; let
. Show the following two statements are equivalent:
(i)
is order preserving and its image is
or a section of
.
(ii)
for all
.
[Hint: Show that each of these conditions implies that
is a section of
; conclude that it must be the section by
.]
(b) If
is a well-ordered set, show that no section of
has the order type of
, nor do two different sections of
have the same order type. [Hint: Given
, there is at most one order-preserving map of
into
whose image is
or a section of
.]
(a) (i) implies (ii). Let
be the set of
such that
, and suppose for some
,
.
is order preserving, therefore, for
,
, and
. Now, since
is
or a section of
,
implies that for all
,
, and there exists
such that
. If
, then
, therefore,
, and
. We conclude that
. By transfinite induction,
, and for all
,
is the smallest element of
not in
.
(ii) implies (i). Let
be the set of
such that
, and suppose for some
,
. If
for some
, then
, contradicting its definition, hence,
for all
, and
. At the same time, by definition, if
, then
,
, implying
. We conclude that
. By transfinite induction,
. This implies that for every
such
,
, and
. Hence,
is order preserving. Further, suppose
. Let us extend
by one element
,
, and assume that for all
,
.
is well-ordered. Consider
such that
, and
. Then
satisfies (ii), and as we already shown,
. This shows that the image of
under
is either the whole
or a section of
by the smallest element not in the image.
(b) By (a), if
is an order preserving map, then it must be given by (ii), and using the principle of the recursive definition (Exercise 1) there is only one such function. So, it must be the identity function (“another” order preserving function), which is not bijective. For two sections, one must be a subsection of the other.