Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms is a set of axioms that allows 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:

1. Isotonicity: Every set is contained in its closure.
2. Idempotence: The closure of the closure of a set is equal to the closure of that set.
3. Preservation of binary unions: The closure of the union of two sets is the union of their closures.
4. Preservation of nullary unions: The closure of the empty set is empty.

In symbols:

1. <math> A \subseteq c(A) \! <math>;
2. <math> c(c(A)) = c(A) \! <math>;
3. <math> c(A \cup B) = c(A) \cup c(B) \! <math>;
4. <math> c(\emptyset) = \emptyset \! <math>.

The closed sets can now be defined as the fixed points of this operator; i.e., all A such that c(A) = 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:

<math> c(A_{1} \cup \cdots \cup A_{n}) = c(A_{1}) \cup \cdots \cup c(A_{n}) \! <math>.

An operator that satisfies only axioms (1) and (2) is called a Moore closure. Moore closure operators are often studied in lattice theory.