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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.
The topic of the book is part of a relatively new field of mathematics crossing between topology and combinatorics, now called topological combinatorics. [2] [3] The starting point of the field, [3] and one of the central inspirations for the book, was a proof that László Lovász published in 1978 of a 1955 conjecture by Martin Kneser, according to which the Kneser graphs +, have no graph ...
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 ...
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 ...
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra . Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory.
If (,) has the homotopy extension property, then the simple inclusion map : is a cofibration.. In fact, if : is a cofibration, then is homeomorphic to its image under .This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.
[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.
In mathematics, the Bing–Borsuk conjecture states that every -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture .