Kuratowski closure axiom

In topology 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:

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.