# Interval and segment

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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
This article was adapted from an original article by BSE-3, L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article