applicability: $\mathbb {R}$

# Definition: Real Intervals

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 $$[a,+ \infty)$$.

Chapters: 1 2
Corollaries: 3 4 5 6 7
Definitions: 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Examples: 25 26
Explanations: 27 28
Lemmas: 29 30 31 32 33
Parts: 34
Proofs: 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
Propositions: 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111
Sections: 112
Subsections: 113 114
Theorems: 115 116 117 118 119 120 121 122 123 124 125

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983