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Absolute neighborhood retracts and shape theory. (English) Zbl 0973.54002

James, I. M. (ed.), History of topology. Amsterdam: Elsevier. 241-269 (1999).
This paper is devoted to the history of two areas of topology that are intimately related. The first of them is the theory of absolute neighborhood retracts (ANR’s) which, according to the author, is the natural environment for homotopy theory and a meeting ground for general and algebraic topology. The second one is the theory of shape, which studies the homotopy-like properties of general spaces by using suitable approximations of these spaces by systems of ANR’s and developing a homotopy theory of systems. Another link between these two theories, and also a justification for considering them together, is the fact that both of them were introduced by the Polish mathematician Karol Borsuk.
The author sketches in the paper the development of the main ideas, starting with the foundational work of Borsuk in the theory of ANR’s presented in his Ph.D. thesis, which he defended in 1930 at the University of Warsaw with the title “On retractions and related sets”. The second fundamental contribution is the foundational paper on the theory of shape “Concerning homotopy properties of compacta” [Fundam. Math. 62, 223-254 (1968; Zbl 0159.24603)]. Another influential article is the author’s paper, written in collaboration with Jack Segal [ibid. 72, 41-59 (1971; Zbl 0222.55017)]. Here the inverse system approach to shape has been introduced, that gives a more categorical description of the theory, complementing Borsuk’s geometric approach. The present paper gives detailed information about the evolution and the significant results of these theories, finishing with the most recent contributions, including applications to fields such as the the theory of continua, dynamical systems, categorical topology and pattern recognition. The paper also provides an accurate bibliography and offers some biographical data on the contributors to both areas. It is an invaluable piece of information for all topologists interested in the theories of ANR’s and shape.
For the entire collection see [Zbl 0922.54003].

MSC:

54-03 History of general topology
54C56 Shape theory in general topology
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
01A60 History of mathematics in the 20th century
55P55 Shape theory
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