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Kazimierz Kuratowski (Polish pronunciation: [kaˈʑimjɛʂ kuraˈtɔfskʲi]; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics .
In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. [1] [2] [3] It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski. [4]
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. If is a graph that contains a subgraph that is a subdivision of or ,, then is known as a Kuratowski subgraph of . [1]
A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map : ℘ ℘ satisfying the following similar requirements: [3] [I1] It preserves the total space : i ( X ) = X {\displaystyle \mathbf {i} (X)=X} ;
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]
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset ) necessarily contains at least one maximal element .
Formally speaking, this embedding was first introduced by Kuratowski, [3] but a very close variation of this embedding appears already in the papers of Fréchet.Those papers make use of the embedding respectively to exhibit as a "universal" separable metric space (it isn't itself separable, hence the scare quotes) [4] and to construct a general metric on by pulling back the metric on a simple ...
The Kuratowski and Ryll-Nardzewski measurable selection theorem says that if X is a Polish space and its Borel σ-algebra, () is the set of nonempty closed subsets of X, (,) is a measurable space, and : is an -weakly measurable map (that is, for every open subset we have {: ()}), then has a selection that is (,)-measurable.