Interval and segment
The simplest sets of points on the line. An interval (open interval) is a set of points on a line lying between two fixed points $ A $ and $ B $, where $ A $ and $ B $ themselves are considered not to belong to the interval. A segment (closed interval) is a set of points between two points $ A $ and $ B $, where $ A $ and $ B $ are included. The terms "interval" and "segment" are also used for the corresponding sets of real numbers: an interval consists of numbers $ x $ satisfying $ a < x < b $, while a segment consists of those $ x $ for which $ a \leq x \leq b $. An interval is denoted by $ ( a , b ) $, or $ \left ] a , b \right [ $, and a segment by $ [ a , b ] $.
The term "interval" is also used in a wider sense for arbitrary connected sets on the line. In this case one considers not only $ ( a , b ) $ but also the infinite, or improper, intervals $ ( - \infty , a ) $, $ ( a , + \infty ) $, $ ( - \infty , + \infty ) $, the segment $ [ a , b ] $, and the half-open intervals $ [ a , b ) $, $ ( a , b ] $, $ ( - \infty , a ] $, $ [ a , + \infty ) $. Here, a round bracket indicates that the appropriate end of the interval is not included, while a square bracket indicates that the end is included.
The notion of an interval in a partially ordered set is more general. An interval $ [ a , b ] $ consists in this setting of all elements $ x $ of the partially ordered set that satisfy $ a \leq x \leq b $. An interval in a partially ordered set that consists of precisely two elements is called a simple or an elementary interval.
Interval and segment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval_and_segment&oldid=47404