Canonical set
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
.
Canonical set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_set&oldid=46197