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Difference between revisions of "Portion"

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''of a set''
 
''of a set''
  
An intersection of the set with an interval in the case of a set on a line, and with an open ball, an open rectangle or an open parallelopipedon in the case of a set in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738401.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738402.png" />. The importance of this concept is based on the following. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738403.png" /> is everywhere dense in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738404.png" /> if every non-empty portion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738405.png" /> contains a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738406.png" />, in other words, if the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738407.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738408.png" /> is nowhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p0738409.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p07384010.png" /> is nowhere dense in any portion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p07384011.png" />, i.e. if there does not exist a portion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p07384012.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073840/p07384013.png" />.
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An intersection of the set with an interval in the case of a set on a line, and with an open ball, an open rectangle or an open parallelopipedon in the case of a set in an $  n $-
 +
dimensional space $  ( n \geq  2 ) $.  
 +
The importance of this concept is based on the following. A set $  A $
 +
is everywhere dense in a set $  B $
 +
if every non-empty portion of $  B $
 +
contains a point of $  A $,  
 +
in other words, if the closure $  \overline{A}\; \supset B $.  
 +
The set $  A $
 +
is nowhere dense in $  B $
 +
if $  A $
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is nowhere dense in any portion of $  B $,  
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i.e. if there does not exist a portion of $  B $
 +
contained in $  \overline{A}\; $.

Latest revision as of 08:07, 6 June 2020


of a set

An intersection of the set with an interval in the case of a set on a line, and with an open ball, an open rectangle or an open parallelopipedon in the case of a set in an $ n $- dimensional space $ ( n \geq 2 ) $. The importance of this concept is based on the following. A set $ A $ is everywhere dense in a set $ B $ if every non-empty portion of $ B $ contains a point of $ A $, in other words, if the closure $ \overline{A}\; \supset B $. The set $ A $ is nowhere dense in $ B $ if $ A $ is nowhere dense in any portion of $ B $, i.e. if there does not exist a portion of $ B $ contained in $ \overline{A}\; $.

How to Cite This Entry:
Portion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Portion&oldid=48247
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article