5.2. Compact and Perfect Sets

Example 5.2.13(a): Properties of the Cantor Set

The Cantor set is compact.
The definition of the Cantor set is as follows: let and define, for each n, the sets A n recursively as Then the Cantor set is given as: 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.

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