applicability: $\mathbb {R}$

Let \(a,b\in\mathbb R\) be real numbers. Using the order relation for real numbers, we call a subset \(I\subseteq\mathbb R\) of real numbers a **real interval**, if it satisfies the following properties:

- \(I:=\{x\in\mathbb R: a\le x \le b\}\). In this case we call \(I\) a
**closed real interval**and denoted it by \([a,b]\). - \(I:=\{x\in\mathbb R: a < x < b\}\). In this case we call \(I\) an
**open real interval**and denoted it by \((a,b)\). Please note that with \(y:=(a+b)/2,~ r=|b-y|=|a-y|\), an open real interval can equivalently be defined as the open ball \(B(y,r)\) in the metric space \((\mathbb R,|~|)\). - \(I:=\{x\in\mathbb R: a < x \le b\}\). In this case we call \(I\) an
**left-open, right-closed real interval**and denoted it by \((a,b]\). - \(I:=\{x\in\mathbb R: a \le x < b\}\). In this case we call \(I\) an
**right-open, left-closed real interval**and denoted it by \([a,b)\).

By allowing either \(a\), or \(b\) to be unbounded, i.e. \(a,b\in\{\mathbb R,+ \infty, -\infty\}\) we define

- \(I:=\{x\in\mathbb R: – \infty < x < b\}\). In this case we call \(I\) a
**left-unbounded, right-open real interval**and denoted it by \((- \infty,b)\). - \(I:=\{x\in\mathbb R: – \infty < x \le b\}\). In this case we call \(I\) a
**left-unbounded, right-closed real interval**and denoted it by \((- \infty,b]\). - \(I:=\{x\in\mathbb R: a < x < + \infty\}\). In this case we call \(I\) a
**right-unbounded, left-open real interval**and denoted it by \((a, + \infty)\). - \(I:=\{x\in\mathbb R: a \le x < + \infty\}\). In this case we call \(I\) a
**right-unbounded, left-closed real interval**and denoted it by \([a,+ \infty)\).

| | | | | created: 2015-02-28 15:34:30 | modified: 2020-07-06 11:47:16 | by: *bookofproofs* | references: [581]

[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983