Section 3.3
71
(b) This follows immediately from part (a) since (0,1]
=
(0,1) U {I}.
25. We employ a concept known
as
stereographic projection.
Imagine the sphere sitting on the Cartesian
plane with south pole at the origin. Any line from the north pole to the plane punctures the sphere at
a unique point and the collection of such lines establishes a onetoone correspondence between the
points of the plane and the sphere except for the north pole. A small modification of this correspon
dence finishes the job. Suppose PO,Pl,P2,
... are the points of the sphere which correspond to the
points (0,0), (1, 0), (2,0),
... in the plane; thus, the line from the north pole to
(n,O)
punctures the
sphere at
Pn
(in particular, Po
=
(0,0». Map the north pole to (0,0), the origin to (1,0), Pl to (2,0),
and so forth and let all other points of the sphere go to the same points
"as
before.
26.
[BB]
We are given that
A
=
{al.a2,
... ,an}
for some
n
E N
and that
B
=
{bl ,b2,b3, .
.
. }. Then
A
U
B
is countably infinite because it is infinite and its elements can be listed as follows:
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 Summer '10
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 Graph Theory, North Pole, South Pole

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