Definition:Real Interval/Empty

Definition
Let $a, b \in \R$.

Let $\left [{a \,.\,.\, b} \right]$, $\left [{a \,.\,.\, b} \right)$, $\left ({a \,.\,.\, b} \right)$ and $\left ({a \,.\,.\, b} \right)$ be real intervals: closed, half-open and open as defined.

When $a > b$:


 * $\left [{a \,.\,.\, b} \right] = \left\{{x \in \R: a \le x \le b}\right\} = \varnothing$


 * $\left [{a \,.\,.\, b} \right) = \left\{{x \in \R: a \le x < b}\right\} = \varnothing$


 * $\left ({a \,.\,.\, b} \right] = \left\{{x \in \R: a < x \le b}\right\} = \varnothing$


 * $\left ({a \,.\,.\, b} \right) = \left\{{x \in \R: a < x < b}\right\} = \varnothing$

When $a = b$:


 * $\left [{a \,.\,.\, b} \right) = \left [{a \,.\,.\, a} \right) = \left\{{x \in \R: a \le x < a}\right\} = \varnothing$


 * $\left ({a \,.\,.\, b} \right] = \left ({a \,.\,.\, a} \right] = \left\{{x \in \R: a < x \le a}\right\} = \varnothing$


 * $\left ({a \,.\,.\, b} \right) = \left ({a \,.\,.\, a} \right) = \left\{{x \in \R: a < x < a}\right\} = \varnothing$

Such empty sets are referred to as empty intervals.