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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]
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 ]
Cauchy space – Concept in general topology and analysis; Convergence space – Generalization of the notion of convergence that is found in general topology; Filters in topology – Use of filters to describe and characterize all basic topological notions and results. Sequential space – Topological space characterized by sequences
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 .
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.
Convex hull (red) of a polygon (yellow). The usual set closure from topology is a closure operator. Other examples include the linear span of a subset of a vector space, the convex hull or affine hull of a subset of a vector space or the lower semicontinuous hull ¯ of a function : {}, where is e.g. a normed space, defined implicitly (¯) = ¯, where is the epigraph of a function .
The Knaster–Kuratowski fan, or "Cantor's teepee" In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point makes it totally disconnected.