# Half-open interval

A half-open interval is an interval which has either a maximum or a minimum element but not both.

If a half-open interval has a minimum $a$ but no maximum, then it is denoted by $[a,b)$, where $b$ is the supremum (least upper bound), or $\infty$ if no supremum exists. Alternatively, $[a,b)$ is the set of all $x$ such that $a \leq x$ and $x < b$.

If a half-open interval has a maximum $b$ but no minimum, then it is denoted by $(a,b]$, where $a$ is the infimum (greatest lower bound), or $-\infty$ if no infimum exists. Alternatively, $(a,b]$ is the set of all $x$ such that $a < x$ and $x \leq b$.

## Examples

$[-1,1)$ is a half-open interval with a minimum but no maximum.

$(-1,1]$ is a half-open interval with a maximum but no minimum.

$[0,\infty)$, the set of nonnegative real numbers, is a half-open interval with no supremum.

$(-\infty,0]$, the set of nonpositive real numbers, is a half-open interval with no infimum.