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Canonical set

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closed, -set

A set of a topological space which is the closure of an open set; in other words, it is the closure of its own interior : . Every closed set contains a maximal -set, namely . The union of two -sets is a -set, but their intersection need not be. A set which is a finite intersection of -sets is called a -set.

A set which is the interior of a closed set is called a canonical open set or -set; in other words, it is a set which is the interior of its own closure: . Every open set is contained in a smallest -set, namely . Open canonical sets can also be defined as complements of closed canonical sets, and vice versa.

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)


Comments

Other terms for canonical set are: regular closed set or closed domain. Canonical open sets are also called regular open sets or open domains.

In the Russian literature denotes the closure of and the interior of . In Western literature these are denoted by and , respectively.

The collection of regular closed sets forms a Boolean algebra under the following operations , and . The same can be done for the collection of regular open sets.

If is a compact Hausdorff space, the Stone space of either one of these algebras is the absolute of .

How to Cite This Entry:
Canonical set. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Canonical_set&oldid=15270
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098