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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 ...
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 .
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.
[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.
Ważewski made important contributions to the theory of ordinary differential equations, partial differential equations, control theory and the theory of analytic spaces. He is most famous for applying the topological concept of retract, introduced by Karol Borsuk, to the study of the solutions of differential equations.
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.
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.