eBook ISBN:  9781470405939 
Product Code:  MEMO/208/979.E 
List Price:  $61.00 
MAA Member Price:  $54.90 
AMS Member Price:  $36.60 
eBook ISBN:  9781470405939 
Product Code:  MEMO/208/979.E 
List Price:  $61.00 
MAA Member Price:  $54.90 
AMS Member Price:  $36.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 208; 2010; 62 ppMSC: 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 compactopen topology. The authors present a complete solution to the topological classification problem for \(\mathcal{H}(M,D)\) as follows. If \(M\) is a onedimensional 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 zerodimensionality

3. Trees and $\mathbb {R}$trees

4. Semicontinuous 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


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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 compactopen topology. The authors present a complete solution to the topological classification problem for \(\mathcal{H}(M,D)\) as follows. If \(M\) is a onedimensional 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 zerodimensionality

3. Trees and $\mathbb {R}$trees

4. Semicontinuous 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