Interval and Segment

the simplest sets of points on a straight line. An interval is the set containing all points of a line between the points A and B but not the points A and B themselves. A segment is the set containing all points of a line between the points A and B as well as these points themselves. The terms “open interval” and “closed interval” are used to denote the corresponding sets of real numbers: an open interval consists of those numbers x that satisfy the inequality a < x < b, while a closed interval consists of those numbers x that satisfy the inequality a ≤ x ≤ b; the open and closed intervals are denoted by (a, b) and [a, b], respectively.

Sometimes the term “interval” is used in a broader sense to denote an arbitrary connected set on a line. In this case, intervals include (finite) open intervals (a, b); infinite open intervals (–∞, a), (a, +∞), (– ∞, + ∞); closed intervals [a, b]; and half-open intervals [a, b), (a, b], (– ∞, a], [a, + ∞). Here the parenthesis denotes that the corresponding endpoint of the interval does not belong to the set being considered, whereas the bracket denotes that it does. For example, (a, b] denotes the set of points x satisfying the inequality a < x ≤ b.