5
Let us conclude these notes with a discussion of the most disconnected set possible –
the
Cantor set
.
Example 20.
Let
C
denote the Cantor set (cf.
§
.244 in [1]). Recall that
C
may be described
as the set of all
even ternary numbers
; that is,
C
⊂
[0
,
1]
consists of numbers
x
whose base3
expansion
[
x
]
3
has only
0
s and
2
s. Let
x
∈
C
, and fix
>
0
. Then, by the Archimedean
property, there is
n
∈
N
with
n
>
1
. Note that
3
n
> n
, and so
>
1
3
n
>
0
. Now, write
x
in
base3,
[
x
]
3
= 0
.x
1
x
2
x
3
. . . x
n

1
x
n
x
n
+1
. . .
This is just shorthand for the expansion
x
=
x
1
3
+
x
2
3
2
+
x
3
3
3
+
· · ·
+
x
n
3
n
+
· · ·
Now, consider the two numbers
x
+
and
x

given by
x
±
=
x
±
1
3
n
.
If
x
n
= 0
then
[
x
+
]
3
= 0
.x
1
x
2
. . . x
n

1
1
x
n
+1
. . .
; if
x
n
= 2
then
[
x

]
3
= 0
.x
1
x
2
. . . x
n

1
1
x
n
+1
. . .
.
In either case,
x
±
do not have even ternary expansions, and so
x
±
/
∈
C
. A similar argu
ment (though with more work due to carrying, as in “carry the
1
. . . ”) shows that
x
±
/
∈
C
in general. Note also that, by choice of
n
,
x
±
∈
B
(
x,
)
.
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 Spring '09
 Koskesh
 Topology, Empty set, Metric space, Topological space, General topology, Clopen set

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