# Erdős Space and Homeomorphism Groups of Manifolds

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*Jan J. Dijkstra; Jan van Mill*

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

# Table of Contents

## Erdos Space and Homeomorphism Groups of Manifolds

- Chapter 1. Introduction 19 free
- Chapter 2. Erdos space and almost zero-dimensionality 513 free
- Chapter 3. Trees and R-trees 715
- Chapter 4. Semi-continuous functions 1119
- Chapter 5. Cohesion 2129
- Chapter 6. Unknotting Lelek functions 2533
- Chapter 7. Extrinsic characterizations of Erdos space 3139
- Chapter 8. Intrinsic characterizations of Erdos space 3947
- Chapter 9. Factoring Erdos space 4957
- Chapter 10. Groups of homeomorphisms 5159
- Bibliography 6169