# Canonical set

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.

#### 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)

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 $X$.