Example:
The Cantor set has length zero, but contains uncountably many
points.
The definition of the Cantor set is as follows: let
and define, for each n, the sets
S n recursively as
- A n = A n-1 \
Then the Cantor set is given as:
-
C =
A n
To be more specific, we have:
- A 0 = [0, 1]
- A 1 = [0, 1] \ (1/3, 2/3)
- A 2 = A 1 \
(1/9, 2/9)
(7/9, 8/9) =
([0,1] \ (1/3, 2/3) ) \ (1/9, 2/9)
(7/9, 8/9)
- ...
That is, at the n-th stage (n > 0) we remove
2 n-1 intervals from each previous set, each
having length 1 / 3 n. Therefore, we will remove
a total length of
from the unit interval [0, 1]. Since we remove a set of total
length 1 from the unit interval, the length of the remaining
Cantor set must be 0.
The Cantor set contains uncountably many points because it is a
perfect set.
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