Open and Closed Sets
Open Sets
An element \(\mathbf{x} \in C \subseteq \mathbb{R}^n\) is called an interior point of \(C\) if there exists and \(\epsilon > 0\) for which:
i.e., there exists a ball centered at \(\mathbf{x}\) that lies entirely in \(C\). The set of all points interior to \(C\) is called the interior of \(C\).
A set \(C\) is open if the interior of \(C\) is equal to \(C\), i.e., every point in \(C\) is an interior point.
Closed Sets
A set \(C \subseteq \mathbb{R}^n\) is closed if its complement \(\mathbb{R}^n \setminus C = \left\{ \mathbf{x} \in \mathbb{R}^n \ | \ \mathbf{x} \notin C \right\}\) is open, i.e., a set is closed if it includes its boundary points. Formally, a set \(X\) is closed if for any convergent sequence in \(X\), its limit point also belongs to \(X\), i.e., if \(\left\{ \mathbf{x}^i \right\} \in X\) and \(\lim_{i \rightarrow \infty} \mathbf{x}^i = \mathbf{x}^0\) then \(\mathbf{x}^0 \in X\). For example, \(X = \mathbb{R}^2\) is a closed set. In contrast, \(X = \left\{x: 0 < x \leq 1 \right\}\) is not a closed set. Note that intersection of closed sets is closed.
Bounded Sets
A set is bounded if it can be enclosed in a large enough (hyper)-sphere or a box. Formally, the set \(X\) is bounded if \(\exists M \geq 0\) such that \(||\mathbf{x}|| \leq M \) for all \( \mathbf{x} \in X\).
Compact Sets
A set that is both bounded and closed is called a compact set.
Examples
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\(x = \mathbb{R}^2\) is closed but not bounded.
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\(X = \left\{ (x, y): \ x^2 + y^2 < 1 \right\}\) is bounded but not closed.
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\(X = \left\{ (x, y): \ x^2 + y^2 \geq 1 \right\}\) is closed but not bounded.
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\(X = \left\{ (x, y): \ x^2 + y^2 \leq 1 \right\}\) is compact.