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''of a topological space''
 
  
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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678101.png" /> defined by the following property: Every non-empty open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678102.png" /> contains a non-empty open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678104.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678105.png" /> is nowhere dense if it is not dense in any non-empty open set.
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{{MSC|54A05|54C05}}
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[[Category:Topology]]
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{{TEX|done}}
  
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A subset $A$ of topological space $X$ is nowhere dense if, for every nonempty open $U\subset X$, the intersection $U\cap A$ is ''not'' dense in $U$. Common equivalent definitions are:
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* For every nonempty open set $U\subset X$, the interior of $U\setminus A$ is not empty.
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* The closure of $A$ has empty interior.
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* The complement of the closure of $A$ is dense.
  
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In an infinite-dimensional Hilbert space, every compact subset is nowhere dense. The same holds for  infinite-dimensional Banach spaces, non-locally-compact Hausdorff topological groups, and products of infinitely many non-compact Hausdorff topological spaces.
  
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====Comments====
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The [[Baire theorem|Baire Category theorem]] asserts that if $X$ is a complete metric space or a locally compact Hausdorff space, then the complement of a countable union of nowhere dense sets is always nonempty.
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Another characterization is: The interior of the closure of a nowhere-dense set is empty. If in a topological product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678106.png" /> infinitely many of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678107.png" /> are non-compact, then each compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678108.png" /> is nowhere dense. A boundary set is the complement of a dense set, i.e. it satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067810/n0678109.png" />. A set whose closure is a boundary set is nowhere dense. A non-empty complete metric space is of the second category, i.e. in it a countable union of nowhere-dense sets is nowhere dense (the Baire category theorem, cf. [[Baire theorem|Baire theorem]]).
 
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , v. Nostrand  (1955) pp. 145</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|AP}}|| A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel  (1984)
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|-
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby, "Measure and category" , Springer  (1971) {{MR|0393403}} {{ZBL|0217.09201}}
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|-
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|valign="top"|{{Ref|Ke}}|| J.L. Kelley, "General topology" , v. Nostrand  (1955) {{MR|0070144}} {{ZBL|0066.1660}}
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|-
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|}

Latest revision as of 19:07, 7 December 2023

2020 Mathematics Subject Classification: Primary: 54A05 Secondary: 54C05 [MSN][ZBL]

A subset $A$ of topological space $X$ is nowhere dense if, for every nonempty open $U\subset X$, the intersection $U\cap A$ is not dense in $U$. Common equivalent definitions are:

  • For every nonempty open set $U\subset X$, the interior of $U\setminus A$ is not empty.
  • The closure of $A$ has empty interior.
  • The complement of the closure of $A$ is dense.

In an infinite-dimensional Hilbert space, every compact subset is nowhere dense. The same holds for infinite-dimensional Banach spaces, non-locally-compact Hausdorff topological groups, and products of infinitely many non-compact Hausdorff topological spaces.

The Baire Category theorem asserts that if $X$ is a complete metric space or a locally compact Hausdorff space, then the complement of a countable union of nowhere dense sets is always nonempty.

References

[AP] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984)
[Ox] J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 Zbl 0217.09201
[Ke] J.L. Kelley, "General topology" , v. Nostrand (1955) MR0070144 Zbl 0066.1660
How to Cite This Entry:
Nowhere-dense set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nowhere-dense_set&oldid=11417
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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