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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 ...
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 system archetype is a pattern of behavior of a system. Systems expressed by circles of causality have therefore similar structure. Identifying a system archetype and finding the leverage enables efficient changes in a system. The basic system archetypes and possible solutions of the problems are mentioned in the Examples section. [1]
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 ...
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.
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.
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 .
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.