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


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
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