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Karol Borsuk (8 May 1905 – 24 January 1982) was a Polish mathematician. His main area of interest was topology . He made significant contributions to shape theory , a term which he coined.
A space is an absolute neighborhood retract for the class , written (), if is in and whenever is a closed subset of a space in , is a neighborhood retract of . Various classes C {\displaystyle {\mathcal {C}}} such as normal spaces have been considered in this definition, but the class M {\displaystyle {\mathcal {M}}} of metrizable spaces ...
Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry is a graduate-level mathematics textbook in topological combinatorics. It describes the use of results in topology , and in particular the Borsuk–Ulam theorem , to prove theorems in combinatorics and discrete geometry .
The Borsuk–Ulam theorem has several equivalent statements in terms of odd functions. Recall that S n {\displaystyle S^{n}} is the n -sphere and B n {\displaystyle B^{n}} is the n -ball : If g : S n → R n {\displaystyle g:S^{n}\to \mathbb {R} ^{n}} is a continuous odd function, then there exists an x ∈ S n {\displaystyle x\in S^{n}} such ...
[7] For all n for fields of revolution — shown by Boris Dekster (1995). [8] The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is no. [9] They claim that their construction shows that n + 1 pieces do not suffice for n = 1325 and for each n > 2014.
Systems analysis is "the process of studying a procedure or business to identify its goal and purposes and create systems and procedures that will efficiently achieve them". Another view sees systems analysis as a problem-solving technique that breaks a system down into its component pieces and analyses how well those parts work and interact to ...
The concept in topology was defined by Karol Borsuk in 1931. [ 2 ] Borsuk's student, Samuel Eilenberg , was with Saunders Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this term to have initially be used.
Constructing the nerve of an open good cover containing 3 sets in the plane.. In topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family.