Infinite dimensional topology and shape theory (Q2776335)

This is a comprehensive survey of the developments in infinite dimensional topology and shape theory that have taken place over the last 35 years. The Hilbert cube \(Q\) (the countable infinite product \(I^\omega\) of the closed unit interval), the separable Hilbert space \(\ell_2\) (known to be homeomorphic to the countable product \(\mathbb R^\omega\) of the real line) and the \(\sigma\)-compact locally convex linear subspaces \(\Sigma\) and \(\sigma\) of \(\ell_2\) are discussed in detail, as well as is the theory of manifolds modeled on them. The author presents the point of view that the \(n\)-dimensional Menger compactum \(\mu^n\) and the \(n\)-dimensional Nöbeling space \(\nu^n\) are the true finite dimensional analogues of \(I^\omega\) and \(\mathbb R^\omega\), respectively. Absolute neighborhood extensors are a major theme in this paper. In particular, they are used for a theory of infinite dimensional topology modulo a complex, which provides a novel unified treatment of the spaces mentioned above. Uncountable products of \(I\) and \(\mathbb R\) (and manifolds modeled on those products) are also discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].