[Understanding Analysis by Stephen Abbott] - [Chapter 3 Basic Topology of R] - [3.2 Open and Closed Sets]

Open and Closed Sets

Definition 3.2.1 (open). A set \(O \subseteq \mathbb{R}\) is open if for all points \(a \in O\) there exists an \(\epsilon\)-neighborhood \(V_\epsilon (a) \subseteq O\)

Example 3.2.2

1. Perhaps the simplest example of an open set is \(\mathbb{R}\) itself.
2. Open interval $$ (c, d) = {x \in \mathbb{R} : c < x < d} $$

Theorem 3.2.3

1. The union of an arbitrary collection of open sets is open.
2. THe intersection of a finite collection of open sets is open

Closed Sets

Definition 3.2.4 (limit point). A point \(x\) is a limit point of a set \(A\) if every \(\epsilon\)-neighborhood \(V_\epsilon (x)\) of \(x\) intersects the set \(A\) at some point other than \(x\).

Theorem 3.2.5. A point \(x\) is a limit point of a set \(A\) if and only if \(x = \lim a_n\) for some sequence \((a_n)\) contained in \(A\) satisfying \(a_n \neq x\) for all \(n \in \mathbb{N}\)

Definition 3.2.6. A point \(a \in A\) is an isolated point of \(A\) if it is not a limit point of \(A\)

Definition 3.2.7 (closed) A set \(F \subseteq \mathbb{R}\) is closed if it contains its limit points.

Theorem 3.2.8. A set \(F \subseteq \mathbb{R}\) is closed if and only if every Cauchy sequence contained in \(F\) has a limit that is also an element of \(F\).

Theorem 3.2.10 (Density of \(\mathbb{Q}\) in \(\mathbb{R}\)). For every \(y \in \mathbb{R}\), there exists a sequence of rational numbers that converges to \(y\).

Closure

Definition 3.2.11. Given a set \(A \subseteq \mathbb{R}\), let \(L\) be the set of all limit points of \(A\). The closure of \(A\) is defined to be \(\overline{A} = A \cup L\)

Theorem 3.2.12 For any \(A \subseteq \mathbb{R}\), the closure \(\overline{A}\) is a closed set and is the smallest closed set containing \(A\).

Complements

Theorem 3.2.13 A set \(O\) is open if and only if \(O^\complement\) is closed. Likewise, a set \(F\) is closed if and only if \(F^\complement\) is open.

Theorem 3.2.14

1. The union of a finite collection of closed sets is closed
2. The intersection of an arbitrary collection of closed sets is closed