Canonical set

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closed, $ \kappa a $- set

A set $ M $ of a topological space which is the closure of an open set; in other words, it is the closure of its own interior $ \langle M\rangle $: $ M = [\langle M\rangle] $. Every closed set $ F $ contains a maximal $ \kappa a $- set, namely $ A = [\langle F \rangle] $. The union of two $ \kappa a $- sets is a $ \kappa a $- set, but their intersection need not be. A set which is a finite intersection of $ \kappa a $- sets is called a $ \pi $- set.

A set which is the interior of a closed set is called a canonical open set or $ \kappa o $- set; in other words, it is a set which is the interior of its own closure: $ M = \langle [M]\rangle $. Every open set $ G $ is contained in a smallest $ \kappa o $- set, namely $ B = \langle [G]\rangle $. Open canonical sets can also be defined as complements of closed canonical sets, and vice versa.


[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)


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 $ [A] $ denotes the closure of $ A $ and $ \langle A\rangle $ the interior of $ A $. In Western literature these are denoted by $ \mathop{\rm Cl} A $ and $ \mathop{\rm Int} A $, respectively.

The collection of regular closed sets forms a Boolean algebra under the following operations $ A \lor B = A \cup B $, $ A \wedge B = \mathop{\rm Cl} ( \mathop{\rm Int} (A \cap B)) $ and $ A ^ \prime = \mathop{\rm Cl} ( \mathop{\rm Int} (X \setminus A) ) $. The same can be done for the collection of regular open sets.

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

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