# Intervals (of the reals), definition

Let \( a, b \in \mathbb{R}^* \) be extended real numbers.

We define the *closed interval* \( [a, b] \) by

We define the *half-open intervals* \( [a, b) \) and \( (a, b] \) by

\( [a, b) := \{x \in \mathbb{R}^* : a \leq x \leq b\}; \) and \( (a, b] := \{x \in \mathbb{R}^* : a \le x \leq b \} \)

And we define the *open interval* \( (a, b) \) by

We call \( a \) the *left endpoint* and \( b \) the *right
endpoint*.

If \( a \) and \( b \) are real numbers (not \( \infty \) or \( -\infty \)), then all intervals above are subsets of the real line.

The real line itself is the open interval \( (-\infty, \infty) \), and the extended real line is the closed interval \( [-\infty, \infty] \).

We sometimes refer to intervals where one endpoint is infinite as being
a *half-infinite* interval, and intervals where both endpoints are
infinite as being a *double-infinite* interval; all other intervals
are *bounded intervals*.