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Homogeneity and rigidity in Erdős spaces. (English) Zbl 1424.54070

Summary: The classical Erdős spaces are obtained as the subspaces of real separable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively.
One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different coordinates it is possible to create a rigid subspace.

MSC:

54F99 Special properties of topological spaces
46A45 Sequence spaces (including Köthe sequence spaces)
54B05 Subspaces in general topology
54D65 Separability of topological spaces
54E50 Complete metric spaces
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
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References:

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