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 and
, where
and
themselves are considered not to belong to the interval. A segment (closed interval) is a set of points between two points
and
, where
and
are included. The terms "interval" and "segment" are also used for the corresponding sets of real numbers: an interval consists of numbers
satisfying
, while a segment consists of those
for which
. An interval is denoted by
, or
, and a segment by
.
The term "interval" is also used in a wider sense for arbitrary connected sets on the line. In this case one considers not only but also the infinite, or improper, intervals
,
,
, the segment
, and the half-open intervals
,
,
,
. 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 consists in this setting of all elements
of the partially ordered set that satisfy
. An interval in a partially ordered set that consists of precisely two elements is called simple.
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.
Interval and segment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval_and_segment&oldid=33603