The Curtis-Schori-West Hyperspace Theorem. Cone Characterization of the Hilbert Cube. Toruńczyk's Approximation Theorem and Applications. Cell-Like Maps and Fine Homotopy Equivalences. Hilbert Space is Homeomorphic to the Countable Infinite Product of Lines. The Estimated Homeomorphism Extension Theorem. The Estimated Homeomorphism Extension Theorem for Compacta in s. Constructing New Homeomorphisms from Old. An Introduction to Infinite-Dimensional Topology. By CldF(X), we denote the space of all closed sets in a space X (including the empty set ) with the Fell topology. The problem of recognizing the topological structure of hyperspaces is a classical problem on the border line of Infinite-dimensional Topology and the Theory of Hyperspaces, studied by many prominent mathematicians: M. Characterization of Finite-Dimensional ANR's and AR's. Various Kinds of Infinite-Dimensionality.ĥ. The Inductive Dimension Functions ind and Ind. The Brouwer Fixed-Point Theorem and Applications. The Michael Selection Theorem and Applications. In the process of proving this result several interesting and useful detours are made. The text is self-contained for readers with a modest knowledge of general topology and linear algebra the necessary background material is collected in chapter 1, or developed as needed.One can look upon this book as a complete and self-contained proof of Toruńczyk's Hilbert cube manifold characterization theorem: a compact ANR X is a manifold modeled on the Hilbert cube if and only if X satisfies the disjoint-cells property. The second part of this book, chapters 7 & 8, is part of geometric topology and is meant for the more advanced mathematician interested in manifolds. Chapter 6 is an introduction to infinite-dimensional topology it uses for the most part geometric methods, and gets to spectacular results fairly quickly. For a student who will go on in geometric or algebraic topology this material is a prerequisite for later work. In chapters 1 - 5, part of the basic material of plane topology, combinatorial topology, dimension theory and ANR theory is presented. 78 (1972) 402-406.The first part of this book is a text for graduate courses in topology. West: 2I is homeomorphic to the Hilbert cube. West: Triangulated infinite-dimensional manifolds. Schori: Topological classification of infinite-dimensional manifolds by homotopy type. CrossRef View in Scopus Google Scholar 6 J.R. New rotation sets in a family of torus homeomorphisms. Vershik Journal of Mathematical Sciences, Vol. The structure of hyperspaces generated by a compact metric space was already studied by several authors. Henderson: Open subsets of Hilbert space. Adic realizations of ergodic actions by homeomorphisms of Markov compacta and ordered Bratteli diagrams A. Henderson: Infinite-dimensional manifolds are open subsets of Hilbert space. hyperspaces of X and the corresponding induced maps on hyperspaces by a. Richard Summerhill: Pseudo-boundaries and pseudo-interiors in Euclidean spaces and topological manifolds. the relationship between homeomorphisms on a compact metric space and their. Schori: Hyperspaces of Peano continua are Hilbert cubes (in preparation). Chapman: All Hilbert cube manifolds are triangulable. Pelczyński: Estimated extension theorem, homogeneous collections and skeletons and their application to topological classification of linear metric spaces. Barit: Small extensions of small homeomorphisms. Chapman: Extending homeomorphisms to Hilbert cube manifolds. Anderson: On sigma-compact subsets of infinite-dimensional spaces. Anderson: On topological infinite deficiency. Anderson: Hilbert space is homeomorphic to the countable infinite product of lines.
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