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Difference between revisions of "Dense set"

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A subset $A$ of topological space $X$ is dense if the closure of $A$ is the entire space $X$. A common alternative definition is:  
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A subset $A$ of a [[topological space]] $X$ is dense for which the [[Closure of a set|closure]] is the entire space $X$ (some authors use the terminology ''everywhere dense''). A common alternative definition is:  
 
* a set $A$ which intersects every nonempty open subset of $X$.  
 
* a set $A$ which intersects every nonempty open subset of $X$.  
  
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A set which is not dense in any non-empty open subset of a topological space $X$ is called [[Nowhere-dense set|nowhere dense]].
 
A set which is not dense in any non-empty open subset of a topological space $X$ is called [[Nowhere-dense set|nowhere dense]].
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A set which consists of limit points is called [[Dense-in-itself set|dense-in-itself]].

Latest revision as of 20:33, 13 December 2017

2020 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

A subset $A$ of a topological space $X$ is dense for which the closure is the entire space $X$ (some authors use the terminology everywhere dense). A common alternative definition is:

  • a set $A$ which intersects every nonempty open subset of $X$.

If $U\subset X$, a set $A\subset X$ is called dense in $U$ if $A\cap U$ is a dense set in the subspace topology of $U$. When $U$ is open this is equivalent to the requirement that the closure (in $X$) of $A$ contains $U$.

A set which is not dense in any non-empty open subset of a topological space $X$ is called nowhere dense.

A set which consists of limit points is called dense-in-itself.

How to Cite This Entry:
Dense set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dense_set&oldid=28109
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article