Compactifications of $\mathbb{N}$ and Polishable subgroups of $S_{\infty}$

Volume 189 / 2006

Todor Tsankov Fundamenta Mathematicae 189 (2006), 269-284 MSC: Primary 54H05, 54H15; Secondary 54F50. DOI: 10.4064/fm189-3-4

Abstract

We study homeomorphism groups of metrizable compactifications of $\mathbb{N}$. All of those groups can be represented as almost zero-dimensional Polishable subgroups of the group $S_\infty$. As a corollary, we show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of $S_\infty$. We prove a sufficient condition for these groups to be one-dimensional and also study their descriptive complexity. In the last section we associate with every Polishable ideal on $\mathbb{N}$ a certain Polishable subgroup of $S_\infty$ which shares its topological dimension and descriptive complexity.

Authors

  • Todor TsankovDepartment of Mathematics 253-37
    California Institute of Technology
    Pasadena, CA 91125, U.S.A.
    e-mail

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