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Erdős Space and Homeomorphism Groups of Manifolds
 
Jan J. Dijkstra Vrije Universiteit, Amsterdam, The Netherlands
Jan van Mill Vrije Universiteit, Amsterdam, The Netherlands
Erdos Space and Homeomorphism Groups of Manifolds
eBook ISBN:  978-1-4704-0593-9
Product Code:  MEMO/208/979.E
List Price: $61.00
MAA Member Price: $54.90
AMS Member Price: $36.60
Erdos Space and Homeomorphism Groups of Manifolds
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Erdős Space and Homeomorphism Groups of Manifolds
Jan J. Dijkstra Vrije Universiteit, Amsterdam, The Netherlands
Jan van Mill Vrije Universiteit, Amsterdam, The Netherlands
eBook ISBN:  978-1-4704-0593-9
Product Code:  MEMO/208/979.E
List Price: $61.00
MAA Member Price: $54.90
AMS Member Price: $36.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2082010; 62 pp
    MSC: Primary 57; 54;

    Let \(M\) be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let \(D\) be an arbitrary countable dense subset of \(M\). Consider the topological group \(\mathcal{H}(M,D)\) which consists of all autohomeomorphisms of \(M\) that map \(D\) onto itself equipped with the compact-open topology. The authors present a complete solution to the topological classification problem for \(\mathcal{H}(M,D)\) as follows. If \(M\) is a one-dimensional topological manifold, then they proved in an earlier paper that \(\mathcal{H}(M,D)\) is homeomorphic to \(\mathbb{Q}^\omega\), the countable power of the space of rational numbers. In all other cases they find in this paper that \(\mathcal{H}(M,D)\) is homeomorphic to the famed Erdős space \(\mathfrak E\), which consists of the vectors in Hilbert space \(\ell^2\) with rational coordinates. They obtain the second result by developing topological characterizations of Erdős space.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Erdős space and almost zero-dimensionality
    • 3. Trees and $\mathbb {R}$-trees
    • 4. Semi-continuous functions
    • 5. Cohesion
    • 6. Unknotting Lelek functions
    • 7. Extrinsic characterizations of Erdős space
    • 8. Intrinsic characterizations of Erdős space
    • 9. Factoring Erdős space
    • 10. Groups of homeomorphisms
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2082010; 62 pp
MSC: Primary 57; 54;

Let \(M\) be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let \(D\) be an arbitrary countable dense subset of \(M\). Consider the topological group \(\mathcal{H}(M,D)\) which consists of all autohomeomorphisms of \(M\) that map \(D\) onto itself equipped with the compact-open topology. The authors present a complete solution to the topological classification problem for \(\mathcal{H}(M,D)\) as follows. If \(M\) is a one-dimensional topological manifold, then they proved in an earlier paper that \(\mathcal{H}(M,D)\) is homeomorphic to \(\mathbb{Q}^\omega\), the countable power of the space of rational numbers. In all other cases they find in this paper that \(\mathcal{H}(M,D)\) is homeomorphic to the famed Erdős space \(\mathfrak E\), which consists of the vectors in Hilbert space \(\ell^2\) with rational coordinates. They obtain the second result by developing topological characterizations of Erdős space.

  • Chapters
  • 1. Introduction
  • 2. Erdős space and almost zero-dimensionality
  • 3. Trees and $\mathbb {R}$-trees
  • 4. Semi-continuous functions
  • 5. Cohesion
  • 6. Unknotting Lelek functions
  • 7. Extrinsic characterizations of Erdős space
  • 8. Intrinsic characterizations of Erdős space
  • 9. Factoring Erdős space
  • 10. Groups of homeomorphisms
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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