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

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