Selected topics in infinite-dimensional topology

Bessaga C., Pełczyński A.

This book is an attempt to give a systematic presentation of results and methods (established mostly in the last ten years) which concern the topology of infinite-dimensional spaces appearing in functional analysis and admitting a local linear structure. Selecting the material we have restricted ourselves to studying and classifying the topological structure of the objects ignoring richer structures (like differentiable structure, Fredholm structure, etc.) and leaving aside any discussion of morphisms between the objects. The permanent rapid progress of the theory resulted that despite our intention we were unable to discuss very recent important results, and ingeneous techniques based on sophisticated topological apparatus. In particular we have not presented J. West’s theorems on factors of the Hilbert cube based on his interior approximation technique, T. A. Chapman’s theory of Hilbert cube manifolds, relationship between Borsuk’s shape and Z-sets discovered by Chapman, the Schori-West theorem stating that the space of closed subsets of the segment [0;1] is homeomorphic to the Hilbert cube. We have also omitted the discussion of homotopy types of general linear groups of Banach spaces.The fundamental results presented in this book are:Theorem of Keller (1931) stating that every infinite-dimensional compact convex subset of the Hilbert space is homeomorphic to the Hilbert cube.Theorem of Kadec and Anderson (1965-66) on topological equivalence of infinite-dimensional separable Frechet spaces;Theorem of Henderson (1969) stating that (under certain assumption on the model) the homotopy equivalence between infinite-dimensional manifolds implies their homeomorphism.