Nowhere-dense set

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