Difference between revisions of "Portion"
From Encyclopedia of Mathematics
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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 | + | 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}\; $. | ||
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
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