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