Kuratowski closure axiom

In topologyrelated branchesmathematics,Kuratowski closure axioms aresetaxioms that allow onedefinetopology onset. They were first introduced by Kazimierz Kuratowski, inslightly different form that applied onlyHausdorff spaces.

In general topology, if X istopological spaceA issubsetX, thenclosureAXdefinedbesmallest closed set containing A, or equivalently,intersectionall closed sets containing A. The closure operator c that assignseach subsetA its closure c(A)thusfunction frompower setXitself. The closure operator satisfiesfollowing axioms:

1. Isotonicity: Every setcontainedits closure.
2. Idempotence: The closure ofclosure ofsetequal toclosurethat set.
3. Preservationbinary unions: The closure ofuniontwo sets isuniontheir closures.
4. Preservationnullary unions: The closure ofempty setempty.

In symbols:

1. ;
2. ;
3. ;
4. .

The closed sets can now be defined asfixed pointsthis operator; i.e., all A such that c(A) = A. Similar setsaxioms existother operators.

Axioms (3)(4) can be generalised (usingproof by mathematical induction) tosingle statement:

• Preservationfinitary unions: The closure ofunionany finite numbersets isuniontheir closures; orsymbols:
• .

An operator that satisfies only axioms (1)(2)calledMoore closure. Moore closure operatorsoften studiedlattice theory.