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Kuratowski Closure AxiomsIn topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which 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 introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator. Definition A topological space is a set with a function -
called the closure operator where is the power set of . The closure operator has to satisfy the following properties - (Isotonicity)
- (Idempotence)
- (Preservation of binary unions)
- (Preservation of nullary unions)
Notes Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the single statement: - (Preservation of finitary unions).
An operator that only satisfies axioms (1) and (2) is called a Moore closure. Moore closure operators are often studied in lattice theory. Recovering topological definitions A function between two topological spaces -
is a called continuous if for all subsets of -
A point is called close to in if is called closed in if . In other words the closed sets of are the fixed points of the closure operator.
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