Ivan Franko National University of Lviv and Jan Kochanowski University of Kielce

## Geometry of Banakh spaces

Abstract: Following Will Brian, we define a metric space X to be Banakh if all nonempty spheres of positive radius r in X have cardinality 2 and diameter 2r. This notion arose from attempts to find a characterization of the real line in the class of metric spaces. Standard examples of Banakh spaces are subgroups of the real line. However, there exist also some exotic examples of Banakh spaces that are quite different from subsets of the real line. In particular, every Hilbert space of density less or equal to the continuum contains a dense Banakh subgroup.

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