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Concept
Kuratowski closure axioms
Summary
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.
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 .
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].
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Trivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.
Closure operator
In mathematics, a closure operator on a set S is a function from the power set of S to itself that satisfies the following conditions for all sets {| border="0" |- | | (cl is extensive), |- | | (cl is increasing), |- | | (cl is idempotent). |} Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families".
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of countability, and all first-countable spaces (especially metric spaces) are sequential. In any topological space if a convergent sequence is contained in a closed set then the limit of that sequence must be contained in as well. This property is known as sequential closure.