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Characterizing Complete Erd\H os Space

  Published:2009-02-01
 Printed: Feb 2009
  • Jan J. Dijkstra
  • Jan van Mill
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Abstract

The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}.
Keywords: Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces
MSC Classifications: 28C10, 46B20, 54F65 show english descriptions Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]
Geometry and structure of normed linear spaces
Topological characterizations of particular spaces
28C10 - Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]
46B20 - Geometry and structure of normed linear spaces
54F65 - Topological characterizations of particular spaces
 

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