Supplementary Exercises*: Well-Ordering: 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
Let
be a set; let
be the collection of all pairs
, where
is a subset of
and
is a well-ordering of
. Define
if
equals a section of
.
(a) Show that
is a strict partial order on
.
(b) Let
be a subcollection of
that is simply ordered by
. Define
to be the union of the sets
, for all
; and define
to be the union of the relations
, for all
. Show that
is a well-ordered set.
(a) No
equals its own section, and if
equals a section of
, which equals a section of
, then
equals a section of
. So, non-reflexivity and transitivity hold.
(b) Let
. Then, for
there exists
. Without loss of generality, suppose
equals a section of
. Then, both points
and
are in
, and at least one of
,
and
holds. Hence, we have comparability. Further,
does not hold in any
, hence,
does not hold, and
is irreflexive. For any three elements
such that
, consider
such that
and
, and suppose that
is a section of
(
equals
or
), then
, implying
and
, hence,
is transitive. To sum up,
is an order on
. For any nonempty subset
of
, let
, and
be such that
. Then, for
,
iff
and
, and the
-smallest element of
is the
-smallest element of
.