The Real And Complex Number Systems

The set of all B qx obtained as x varies over all elements of A is a countable collection
of open sets which covers A. To get a countable subcollection of F which covers A, we
simply correlate to each set B qx one of the sets G of F which contained B qx . This
complete the proof.
3.35 Refer to exercise 3.32. If A is dense in S and B is open in S, prove that
B clA B, where clA B means the closure of A B.
Hint. Exercise 3.13.
Proof: Since A is dense in S and B is open in S, A S and S B B. Then
B S B
A B, B is open in S
clA B
by exercise 3.13.
3.36 Refer to exercise 3.32. If each of A and B is dense in S and if B is open in S, prove
that A BisdenseinS.
Proof: Since
clA B, B is open
clA B by exercise 3.13
S B since A is dense in S
B since B is open in S
then
clA B B
which implies
clA B S
since B is dense in S.
3.37 Given two metric spaces S 1 , d 1 and S 2 , d 2 , ametric for the Cartesian
product S 1 S 2 can be constructed from d 1 d 2 in may ways. For example, if x x 1 , x 2
and y y 1 , y 2 are in S 1 S 2 , let x, y d 1 x 1 , y 1 d 2 x 2 , y 2 . Prove that is a
metric for S 1 S 2 and construct further examples.
Proof: In order to show that isametricforS 1 S 2 , we consider the following four
steps.
(1) For x x 1 , x 2 S 1 S 2 , x, x d 1 x 1 , x 1 d 2 x 2 , x 2 0 0 0.
(2) For x y, x, y d 1 x 1 , y 1 d 2 x 2 , y 2 0 since if x, y 0, then x 1 y 1
and x 2 y 2 .
(3) For x, y S 1 S 2 ,
x, y d 1 x 1 , y 1 d 2 x 2 , y 2
d 1 y 1 , x 1 d 2 y 2 , x 2
y, x.
(4) For x, y, z S 1 S 2 ,
x, y d 1 x 1 , y 1 d 2 x 2 , y 2
d 1 x 1 , z 1 d 1 z 1 , y 1 d 2 x 2 , z 2 d 2 z 2 , y 2
d 1 x 1 , z 1 d 2 x 2 , z 2 d 1 z 1 , y 1 d 2 z 2 , y 2
x, z z, y.