Example 5.2.13(a): Properties of the Cantor Set
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The Cantor set is compact.
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The definition of the Cantor set is as follows: let
and define, for each
n, the sets
A n
recursively as
-
A n = A n-1 /
Then the Cantor set is given as:
- C =
A n
Each set
is open. Since
A 0 is closed, the sets
A n are all closed as well, which can be shown by
induction. Also, each set
A n
is a subset of
A 0,
so that all sets
A n
are bounded.
Hence, C is the intersection of closed, bounded sets, and
therefore C is also closed and bounded. But then
C is compact.