An open interval is an interval which has neither a maximum nor a minimum element. Open intervals are denoted by where is the infimum (greatest lower bound) and is the supremum (least upper bound). Alternatively, an open interval is the set of all such that satisfies both of the inequalities and .
In an open interval, it is possible that either the infimum or the supremum , or both, do not exist. If the infimum is nonexistent, the value of is written as ; if the supremum is nonexistent, the value of is written as . The corresponding inequality is always considered true in these cases, since and by definition.
Every open interval is an open set.
, the set of all real numbers, is an open interval with neither an upper bound nor a lower bound.
, the set of positive real numbers, is an open interval with a lower bound but no upper bound.
, the set of negative real numbers, is an open interval with an upper bound but no lower bound.
, the set of real numbers strictly between and , 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.
This article is a stub. Help us out by.