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The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR.
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.
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.
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 .
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.
Karol Borsuk, Polish mathematician; his main area of interest was topology; he introduced the theory of absolute retracts (ARs) and absolute neighborhood retracts (ANRs), and the cohomotopy groups, later called Borsuk–Spanier cohomotopy groups; he also founded shape theory; Borsuk's conjecture, Borsuk-Ulam theorem. [79] [80]
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.
X deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.) For any path-connected space Y, any two maps f,g: X → Y are homotopic. For any nonempty space Y, any map f: Y → X is null-homotopic. The cone on a space X is always contractible. Therefore any space can be ...