Interactive Real Analysis

• 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 5.2.10(a):

Find a perfect set. Find a closed set that is not perfect. Find a compact set that is not perfect. Find an unbounded closed set that is not perfect. Find a closed set that is neither compact nor perfect.

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• A perfect set needs to be closed, such as the closed interval [a, b]. In fact, every point in that interval [a, b] is an accumulation point, so that the set [a, b] is a perfect set.
• The simplest closed set is a singleton { b }.The element b in then set { b } is not an accumulation point, so the set { b } is closed but not perfect.
• The set { b } from above is also compact, being closed an bounded. Hence, it is compact but not perfect.
• The set {-1} [0, ) is closed, unbounded, but not perfect, because the element -1 is not an accumulation point of the set.
• The set {-1} [0, ) from above is closed, not perfect, and also not compact, because it is unbounded.