Examples
Closed intervals are perfect sets except for the singleton sets \([a,a]\).
The Cantor set is also a perfect set. This is because \(C = \bigcap_{n=0}^{\infty} C_n\) where each \(C_n\) is a finite union of closed intervals and we know. By theorem 3.2.14 which states that the union of a finite collection of closed sets is closed, this means that \(C_n\) is closed and so \(C\) is closed as well. What about isolated points? Let \(x \in C\) be arbitrary. To show that it’s not isolated, we need to show that it’s a limit point. This mean that there is some sequence \((x_n)\) such that \((x_n) \rightarrow x\) and \(x\) is not a term in the sequence. (Exercise 3.4.3 will contain the rest of this proof: TODO).
Other Definitions and Properties
- Open Sets
- Limit Points
- Closed Sets
- Compact Sets
- Closure
- Absolute Value Function
- Sequences, Subsequences and Convergence
- Series and Series Convergence
- Show the Limit Template
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