Kuratowski closure axiom
and related branches of mathematics
, the Kuratowski closure axioms
are a set of axioms
that allow one to define a topology
on a set
They were first introduced by Kazimierz Kuratowski
, in a slightly different form that applied only to Hausdorff spaces.
In general topology, if X is a topological space and A is a subset of X, then the closure of A in X is defined to be the smallest closed set containing A, or equivalently, the intersection of all closed sets containing A.
The closure operator c that assigns to each subset of A its closure c(A) is thus a function from the power set of X to itself.
The closure operator satisfies the following axioms:
- Isotonicity: Every set is contained in its closure.
- Idempotence: The closure of the closure of a set is equal to the closure of that set.
- Preservation of binary unions: The closure of the union of two sets is the union of their closures.
- Preservation of nullary unions: The closure of the empty set is empty.
The closed sets can now be defined as the fixed points of this operator; i.e., all A
such that c
) = A
Similar sets of axioms exist for other operators.
Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the single statement:
- Preservation of finitary unions: The closure of the union of any finite number of sets is the union of their closures; or in symbols:
An operator that satisfies only axioms (1) and (2) is called a Moore closure
Moore closure operators are often studied in lattice theory