Section 3: Problem 9 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
Check that the dictionary order is an order relation.
CnRT clearly hold. Indeed, for
,
and
, where
, where
is an ordered set, the C property of
implies that either
, in which case either
(
) or
(
) or
(
), or
(
), or
(
). Further, the nR property of
implies that
, hence,
. Finally, if
, then either
(in
), implying
, or
(why only these two cases?) and
, again implying
.