Topics on Continua

Topics on Continua

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ISBN 9780849337383
Cat# DK6026
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Features

  • Offers the first detailed, systematic treatment of Jones's set function T, homogeneous continua, and n-fold hyperspaces available in book form
  • Presents theorems not found in print anywhere else
  • Includes many illustrations that help clarify the definitions and proofs
  • Breaks new ground with important unsolved problems and unique problems for research
  • Summary

    Specialized as it might be, continuum theory is one of the most intriguing areas in mathematics. However, despite being popular journal fare, few books have thoroughly explored this interesting aspect of topology.

    In Topics on Continua, Sergio Macías, one of the field’s leading scholars, presents four of his favorite continuum topics: inverse limits, Jones’s set function T, homogenous continua, and n-fold hyperspaces, and in doing so, presents the most complete set of theorems and proofs ever contained in a single topology volume. Many of the results presented have previously appeared only in research papers, and some appear here for the first time.

    After building the requisite background and exploring the inverse limits of continua, the discussions focus on Professor Jones's set function T and continua for which T is continuous. An introduction to topological groups and group actions lead to a proof of Effros's Theorem, followed by a presentation of two decomposition theorems. The author then offers an in-depth study of n-fold hyperspaces. This includes their general properties, conditions that allow points of n-fold symmetric products to be arcwise accessible from their complement, points that arcwise disconnect the n-fold hyperspaces, the n-fold hyperspaces of graphs, and theorems relating n-fold hyperspaces and cones. The concluding chapter presents a series of open questions on each topic discussed in the book.

    With more than a decade of teaching experience, Macías is able to put forth exceptionally cogent discussions that not only give beginning mathematicians a strong grounding in continuum theory, but also form an authoritative, single-source guide through some of topology's most captivating facets.

    Table of Contents

    PRELIMINARIES
    Product Topology
    Continuous Decompositions
    Homotopy and Fundamental Group
    Geometric Complexes and Polyhedra
    Complete Metric Spaces
    Compacta
    Continua
    Hyperspaces
    References
    INVERSE LIMITS AND RELATED TOPICS
    Inverse Limits
    Inverse Limits and the Cantor Set
    Inverse Limits and Other Operations
    Chainable Continua
    Circularly Chainable and P–like Continua
    Universal and A–H Essential Maps
    References
    JONES’S SET FUNCTION T
    The Set Function T
    Continuity of T
    Applications
    References
    A THEOREM OF E. G. EFFROS
    Topological Groups
    Group Actions and a Theorem of Effros
    References
    DECOMPOSITION THEOREMS
    Jones’s Theorem
    Detour to Covering Spaces
    Rogers’s Theorem
    Case and Minc–Rogers Continua
    Covering Spaces of Some Homogeneous Continua
    References
    n–FOLD HYPERSPACES
    General Properties
    Unicoherence
    Aposyndesis
    Arcwise Accessibility
    Points that Arcwise Disconnect
    C*n–smoothness
    Retractions
    Graphs
    Cones
    References
    QUESTIONS
    Inverse Limits
    The Set Function T
    Homogeneous Continua
    n–fold Hyperspaces
    References
    Index

    Editorial Reviews

    “The book consists of four expository papers … topics: inverse limits, Jones’ set function T, homogeneous continua and n-fold hyperspaces. … It includes a quick survey of some topics on metric spaces and general topology. … It presents standard results and shows that the operation of taking inverse limits commutes with the operation of taking finite products, cones and hyperspaces. … problems to the topics of the book. …”
    — Alejandro Illanes, Mexico, in Zentralblatt MATH, Vol. 1081, 2006/07

     
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