Namespaces
Variants
Actions

Nowhere-dense set

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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=54755
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article