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

Mentioned in:

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

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References

Bibliography

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