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Difference between revisions of "Dense set"

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The same as an [[Everywhere-dense set|everywhere-dense set]]. More generally, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310701.png" /> is called dense in an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310702.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310703.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310704.png" /> is contained in the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310705.png" /> or, which is the same, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310706.png" /> is everywhere dense in the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310707.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310708.png" /> is not dense in any non-empty open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d0310709.png" />, it is a [[Nowhere-dense set|nowhere-dense set]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031070/d03107010.png" />.
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{{MSC|54A05}}
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[[Category:Topology]]
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{{TEX|done}}
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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:
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* a set $A$ which intersects every nonempty open subset of $X$.  
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If $U\subset X$, a set $A\subset X$ is called ''dense'' in $U$ if $A\cap U$
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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 $U$.
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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]].
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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=16528
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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