In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator. Definition Kuratowski closure operators and weakenings Let be an arbitrary set and its power set. A Kuratowski closure operator is a unary operation with the following properties: A consequence of preserving binary unions is the following condition: In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity): then it is easy to see that axioms [K4'] and [K4''] together are equivalent to [K4] (see the next-to-last paragraph of Proof 2 below). includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all , . He refers to topological spaces which satisfy all five axioms as T1-spaces in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T1-spaces via the usual correspondence (see below). If requirement [K3] is omitted, then the axioms define a Čech closure operator. If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator. A pair is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by . Alternative axiomatizations The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin: Axioms [K1]–[K4] can be derived as a consequence of this requirement: Choose . Then , or . This immediately implies [K1]. Choose an arbitrary and . Then, applying axiom [K1], , implying [K2]. Choose and an arbitrary . Then, applying axiom [K1], , which is [K3]. Choose arbitrary . Applying axioms [K1]–[K3], one derives [K4]. Alternatively, had proposed a weaker axiom that only entails [K2]–[K4]: Requirement [K1] is independent of [M] : indeed, if , the operator defined by the constant assignment satisfies [M] but does not preserve the empty set, since . Notice that, by definition, any operator satisfying [M] is a Moore closure operator. A more symmetric alternative