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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}\; $.

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This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article