Namespaces
Variants
Actions

Nowhere-dense set

From Encyclopedia of Mathematics
Revision as of 16:55, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

of a topological space

A set defined by the following property: Every non-empty open set contains a non-empty open set such that . In other words, is nowhere dense if it is not dense in any non-empty open set.


Comments

Another characterization is: The interior of the closure of a nowhere-dense set is empty. If in a topological product infinitely many of the spaces are non-compact, then each compact subset of is nowhere dense. A boundary set is the complement of a dense set, i.e. it satisfies . 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).

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[a2] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 145
How to Cite This Entry:
Nowhere-dense set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nowhere-dense_set&oldid=28105
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article