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Difference between revisions of "Interval and segment"

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''BSE-3''
 
''BSE-3''
  
The notion of an interval in a partially ordered set is more general. An interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209025.png" /> consists in this setting of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209026.png" /> of the [[Partially ordered set|partially ordered set]] that satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209027.png" />. An interval in a partially ordered set that consists of precisely two elements is called simple.
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The notion of an interval in a [[partially ordered set]] is more general. An interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209025.png" /> consists in this setting of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209026.png" /> of the partially ordered set that satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209027.png" />. An interval in a partially ordered set that consists of precisely two elements is called a ''simple'' or an ''[[elementary interval]]''.
  
 
''L.A. Skornyakov''
 
''L.A. Skornyakov''
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====Comments====
 
====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, open]]; [[Interval, closed|Interval, closed]].
 
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, open]]; [[Interval, closed|Interval, closed]].
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[[Category:Order, lattices, ordered algebraic structures]]

Revision as of 22:02, 12 October 2014

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 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=14087
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