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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.
In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. [1]
Kuratowski proved the Kuratowski-Zorn lemma (often called just Zorn's lemma) in 1922. [6] This result has important connections to many basic theorems. Zorn gave its application in 1935. [7] Kuratowski implemented many concepts in set theory and topology. In many cases, Kuratowski established new terminologies and symbolisms.
For example, in Kazimierz Kuratowski's well-known textbook on point-set topology, a topological space is defined as a set together with a certain type of "closure operator," and all other concepts are derived therefrom. [2]
Every Kuratowski subgraph is a special case of a minor of the same type, and while the reverse is not true, it is not difficult to find a Kuratowski subgraph (of one type or the other) from one of these two forbidden minors; therefore, these two theorems are equivalent. [12] An extension is the Robertson–Seymour theorem.
The definition of a point of closure of a set is closely related to the definition of a limit point of a set.The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...
In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902, [ 1 ] the concept was popularized in texts by Felix Hausdorff [ 2 ] and Kazimierz Kuratowski . [ 3 ]
The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz. [ 1 ] The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem .