Published Jun 28, 2021
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Definition 3.4.1 A set \(P \subseteq \mathbb{R}\) is perfect if it is closed and contains no isolated points.
Definition 3.4.3 A nonempty perfect set is uncountable.
Definition 3.4.4 Two nonempty sets \(A, B \subseteq \mathbb{R}\) are separated if \(\overline{A} \cap B\) and \(A \cap \overline{B}\) are both empty. A set \(E \subseteq \mathbb{R}\) is disconnected if it can be written as \(E = A \cup B\), where \(A\) nad \(B\) are nonempty separated sets.
A set that is not disconnected is called a connected set
Theorem 3.4.6 A set \(E \subseteq \mathbb{R}\) is connected if and only if, for all nonempty disjoint sets \(A\) and \(B\) satisfying \(E = A \cup B\), there always exists a convergent sequence \((x_n) \to x\) with \((x_n)\) contained in one of \(A\) or \(B\), and x an element of the other
Theorem 3.4.7 A set \(E \subseteq \mathbb{R}\) is connected if and only if whenever \(a < c < b\) with \(a, b \in E\), it follows that \(c \in E\) as well.