View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520902.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520904.png" /> themselves are considered not to belong to the interval. A segment (closed interval) is a set of points between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520906.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520908.png" /> are included. The terms  "interval"  and  "segment"  are also used for the corresponding sets of real numbers: an interval consists of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i0520909.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209010.png" />, while a segment consists of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209011.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209012.png" />. An interval is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209013.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209014.png" />, and a segment by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209015.png" />.

The term  "interval"  is also used in a wider sense for arbitrary connected sets on the line. In this case one considers not only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209016.png" /> but also the infinite, or improper, intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209019.png" />, the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209020.png" />, and the half-open intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052090/i05209024.png" />. 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 <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''

====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]].

[[Category:Order, lattices, ordered algebraic structures]]