Normality on topological groups

Abstract

It is a well known fact that every topological group which satisfies a midl separation axiom like beint T0, is automatically Hausdorff and completely regular, thus, a Tychonoff space. Further separation axioms do not hold in general. For instance, the topological produt of uncountable many copies of the discrete group of integer numbers, say ZR is not normal. Clearly it is a topological Abelian Hausdorff group, with the operation defined pointwise and the product topology t. With this example in mind, one can ask, are there "many non-normal" groups? Markov asked in 1945 whether every uncountable abstract group admits a non-normal group topology. Van Douwen in 1990 asked if every Abelian group endowed with the weak topology corresponding to the family of all its homomorphisms in the unit circle of the complex plane should be normal. Here we prove that the above group ZR endowed with its Bohr topology tb is non-normal either, and obtain that all group topologies on ZR which lie between tb and the original one t are also non-normal. In fact, every compatible topology for this group lacks normality and we raise the general question about the "normality behaviour" of compatible group topologies.