Open interval

An open interval is an interval which has neither a maximum nor a minimum element. Open intervals are denoted by $(a,b)$ where $a$ is the infimum (greatest lower bound) and $b$ is the supremum (least upper bound). Alternatively, an open interval $(a,b)$ is the set of all $x$ such that $x$ satisfies both of the inequalities $a < x$ and $x < b$.

In an open interval, it is possible that either $a$ or $b$, or both, do not exist. If $a$ is nonexistent, the value of $a$ is written as $-\infty$; if b is nonexistent, the value of $b$ is written as $\infty$. The corresponding inequality is always considered true in these cases, since $-\infty < x$ and $x < +\infty$ by definition.

Every open interval is an open set.

Examples

$\mathbb R$, the set of all real numbers, is an open interval with neither an upper bound nor a lower bound.

$(0, + \infty)$, the set of positive real numbers, is an open interval with a lower bound but no upper bound.

$(-\infty, 0)$, the set of negative real numbers, is an open interval with an upper bound but no lower bound.

$(-1, 1)$, the set of real numbers strictly between $-1$ and $1$, is an open interval with both an upper bound and a lower bound.

The empty set, having no elements and therefore neither a maximum nor a minimum, is considered an open interval.

See also

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