# Nowhere-dense set

2010 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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article