The closed interval of real numbers with lower bound \(a\) and upper bound \(b\) is defined as the set of all real numbers, \(x\), such that \(a \leq x \leq b\).

Similarly, the open interval of real numbers with lower bound \(a\) and upper bound \(b\) is defined as the set of all real numbers, \(x\), such that \(a \lt x \lt b\).

The notation \([a , b]\) is often used to denote the closed interval of real numbers with lower bound \(a\) and upper bound \(b\). That is:

\[ [a,b] = \{x \in \mathbb{R} : a \leq x \leq b \} \]

Simlarly, the notation \((a , b)\) is often used to denote the open interval of real numbers with lower bound \(a\) and upper bound \(b\). That is:

\[ (a,b) = \{x \in \mathbb{R} : a \lt x \lt b \} \]

It is sometimes necessary to work with intervals that are said to be either half-open or half-closed. In such situations, the following notation is commonly used:

\[ (a,b] = \{x \in \mathbb{R} : a \lt x \leq b \} \]

\[ [a,b) = \{x \in \mathbb{R} : a \leq x \lt b \} \]