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3
Example 12.
In Example 8, we saw that
C
1
= [0
,
1]
in the metric space
[0
,
1]
∪
(2
,
3)
. Fol
lowing Lemma 9, this means that for any point
x
∈
[0
,
1]
,
C
x
= [0
,
1]
(indeed, this is
easy to show directly). On the other hand, if
y
∈
(2
,
3)
, the reader can readily verify that
C
y
= (2
,
3)
. Hence, the metric space
[0
,
1]
∪
(2
,
3)
has two connected components:
[0
,
1]
and
(2
,
3)
.
Example 13.
Consider the metric space
Q
equipped with the usual metric
d
(
q,p
)=

q

p

.
Let
q
0
be any nonzero rational number. By the Archimedean property of
Q
, there exists a
natural number
n
so that
n
·
q
2
0
>
1
. Since
n
2
+1
>n
, we therefore have
(
n
2
+ 1)
·
q
2
0
>
1
.
Now, de±ne
A
=
{
p
∈
Q
;
p
2
>
1
1+
n
2
}
and
B
=
{
p
∈
Q
;
p
2
<
1
1+
n
2
}
. By de±nition,
q
0
∈
A
.
Note,
1+
n
2
is not a perfect square (since
n>
0
here), and therefore (following standard
proofs like the one in the ±rst lecture)
1
1+
n
2
is not the square of any rational number. It
follows that
Q
=
A
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This note was uploaded on 05/06/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.
 Spring '09
 Koskesh

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