Continuous maps in the Bohr Topology

Abstract

[EN] The Bohr topology of an Abelian group G is the initial topology on G with respect to the family of all homomorphisms of G into the circle group. The group G equipped with the Bohr topology is denoted by G#. It was an open question of van Douwen whether for any two discrete abelian groups G and H of the same cardinality the topological spaces G# and H# are homeomorphic. A negative solution to van Douwen's problem was given independently by Kunen and by Watson and the authot. In both cases infinite dimensional vector spaces Vp over the finite field Zp were used to show that there is no homeomorphism between V#p and V#q for p not = q and |Vp| = |Vq|. More precisely, it was shown that every continuous map V#p -> V#q is more constant on an infinite subset of Vp hence cannot be a homeomorphism. Motivated by this phenomenon we establish in this paper the "typical" behavior of a continuous mpa f:V#2 -> H# (and discuss without proofs the more gnereal case f:V#p ->H#). The specific choice of p=2 permits to consider V2 as the set of all finite subsets of an infinite set B (the base of V2). A special attention wil be paid to the restriction of f to the doubletons and the four element subsets of B.