Glossary of Topology - K

K

Kolmogorov axiom

See T₀.

Kuratowski closure axioms

The Kuratowski closure axioms is a set of axioms satisfied by the function which takes each subset of X to its closure:

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.

If c is a function from the power set of X to itself, then c is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on X by declaring the closed sets to be the fixed points of this operator, i.e. a set A is closed if and only if c(A) = A.