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,[1] and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro,[2] among others.
A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.[3]
^Kuratowski (1922).
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^Cite error: The named reference :2 was invoked but never defined (see the help page).
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In topology and related branches of mathematics, the Kuratowskiclosureaxioms are a set of axioms that can be used to define a topological structure on...
concepts bearing Kuratowski's name include Kuratowski's theorem, Kuratowskiclosureaxioms, Kuratowski-Zorn lemma and Kuratowski's intersection theorem...
points and satisfy the correct axioms. Another way to define a topological space is by using the Kuratowskiclosureaxioms, which define the closed sets...
isometry is surjective. Kolmogorov axiom See T0. Kuratowskiclosureaxioms The Kuratowskiclosureaxioms is a set of axioms satisfied by the function which...
given set Kuratowskiclosureaxioms – mathematical conceptPages displaying wikidata descriptions as a fallback Preclosure operator – Closure operator Diatta...
the power set of X . {\displaystyle X.} One has the following Kuratowskiclosureaxioms: A ⊆ cl ( A ) {\displaystyle A\subseteq \operatorname {cl} (A)}...
itself, are examples of closure operators since they satisfy all of the Kuratowskiclosureaxioms. As a result, the upper closure of a set is equal to the...
of braces from the canonical Kuratowski definition (a,b) = {{a},{a,b}}. This was actually the original form of the axiom in von Neumann's axiomatization...
topological closure cl X A {\displaystyle \operatorname {cl} _{X}A} satisfies the Kuratowskiclosureaxioms. Conversely, for any closure operator A ↦...
spaces which fulfilled the Kuratowskiclosureaxioms up to the axiom of idempotence. These spaces are often also called closure spaces, and Hausdorff used...
idempotent: the last containment may be strict. Thus sequential closure is not a (Kuratowski) closure operator. A set S {\displaystyle S} is sequentially closed...
provided two axioms for independence, and defined any structure adhering to these axioms to be "matroids". His key observation was that these axioms provide...
first axiom is *12.1; the second is *12.11. To quote Wiener the second axiom *12.11 "is involved only in the theory of relations". Both axioms, however...
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The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma (Kelley 1955:33). The Hausdorff maximal principle states that, in...
all the axioms of a metric are satisfied except that the distance between identical points is not necessarily zero. In other words, the axioms for a metametric...
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