Section 6: 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
(a) Let
. Show there is a bijection of
with the cartesian product
, where
is the two-element set
.
(b) Show that if
is finite, then
is finite.
(a) Let
where
iff
. In other words, the sequence of
and
tells which numbers are in the subset and which are not.
is clearly injective and surjective.
(b) Let
be bijective. For
, let
where
iff
(similar to (a)). Then,
is a bijective correspondence between
and
, which, according to Corollary 6.8, is finite. Hence, there is a bijective correspondence
of
with a finite section of the positive numbers, and
is a bijective correspondence of
with the same section of the positive numbers.