Difference between revisions of "Nowhere-dense set"
From Encyclopedia of Mathematics
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* The complement of the closure of $A$ is dense. | * The complement of the closure of $A$ is dense. | ||
| − | In | + | 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 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. | 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. | ||
Revision as of 06:51, 23 September 2012
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 0217.09201 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=28115
Nowhere-dense set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nowhere-dense_set&oldid=28115
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article