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History of shape theory and its application to general topology. (English) Zbl 0996.54002

Aull, C. E. (ed.) et al., Handbook of the history of general topology. Volume 3. Papers from the special session of the American Mathematical Society meeting, San Antonio, TX, USA, 1993. Dordrecht: Kluwer Academic Publishers. Hist. Topol. 3, 1145-1177 (2001).
As the title promises, the authors treat very competently (as can be expected according to the names of the authors) the history of shape, in particular of strong shape theory and its applications to general topology. This implies, that the various and interesting connections to algebraic topology are only very shortly sketched. Apart from the different definitions of shape categories which appeared during several decades, beginning with E. Lima’s first approach to stable shape, over Borsuk’s definition to more recent developments, the reader finds a good introduction to strong shape, the theory of movable compacta, FANRs, fixed point theory and so-called hyperspaces. Moreover, the applications to dynamical systems are treated. The paper does not contain any proofs, but is a very good source for suggestions for further reading. The list of references contains 285 items. For everybody, interested in shape theory this article constitutes doubtlessly a basic and indispensable tool for his work.
For the entire collection see [Zbl 0980.00038].

MSC:

54-03 History of general topology
54C56 Shape theory in general topology
57N25 Shapes (aspects of topological manifolds)
55P55 Shape theory
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