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Difference between revisions of "Interval, open"

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''open segment''
 
''open segment''
  
The set of points contained in between two given constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521202.png" />, i.e. satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521203.png" />. An open interval does not contain the end points, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521204.png" />, as distinct from the (closed) segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521205.png" /> (a closed interval), which contains the end points, i.e. consists of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521207.png" />.
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The set of points contained in between two given constants $a$ and $b$, i.e. satisfying the condition $a<x<b$. An open interval does not contain the end points, and is denoted by $(a,b)$, as distinct from the (closed) segment $[a,b]$ (a closed interval), which contains the end points, i.e. consists of the points $x$, $a\leq x\leq b$.
  
  
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See also [[Interval, closed|Interval, closed]]; [[Interval and segment|Interval and segment]].
 
See also [[Interval, closed|Interval, closed]]; [[Interval and segment|Interval and segment]].
  
Many authors use the (increasingly popular) alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521208.png" /> to denote the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052120/i0521209.png" />.
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Many authors use the (increasingly popular) alternative $]a,b[$ to denote the open interval $a<x<b$.

Latest revision as of 13:30, 9 April 2014

open segment

The set of points contained in between two given constants $a$ and $b$, i.e. satisfying the condition $a<x<b$. An open interval does not contain the end points, and is denoted by $(a,b)$, as distinct from the (closed) segment $[a,b]$ (a closed interval), which contains the end points, i.e. consists of the points $x$, $a\leq x\leq b$.


Comments

See also Interval, closed; Interval and segment.

Many authors use the (increasingly popular) alternative $]a,b[$ to denote the open interval $a<x<b$.

How to Cite This Entry:
Interval, open. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval,_open&oldid=31443
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article