• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 5.1. Open and Closed Sets
• 5.2. Compact and Perfect Sets
• 5.3. Connected and Disconnected Sets
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

5.2. Compact and Perfect Sets

Example 5.2.13(a): Properties of the Cantor Set

The definition of the Cantor set is as follows: let

• A ₀ = [0, 1]

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.

Interactive Real Analysis, ver. 2.0.1(c)
(c) 1994-2018, Bert G. Wachsmuth
Page last modified: Apr 25, 2024