# 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.

*BSE-3*

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*.

*L.A. Skornyakov*

#### Comments

In English the term "segment" is not often used, except in specifically geometrical contexts; the normal terms are "open interval" and "closed interval" , cf. also Interval, open; Interval, closed.

**How to Cite This Entry:**

Interval and segment.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Interval_and_segment&oldid=47404