© 1996, Topology Atlas

The motivation of studying the problem of my paper "Regular Semitopological Groups of every Countable Sequential Order" (Topology and its Applications, 58 (1994), pp.145-149) was a question proposed by Nyikos, namely whether there exists regular topological semigroups of sequential order strictly between $1$ and $\omega_1$ and in that paper I proved that this is so, and in fact I proved that there exist countable regular semitopological Abelian groups of every countable sequential order. An analogous result on semitopological groups was proved previously in a paper by S. Dolecki and myself. A more natural problem of this type is whether similar properties hold for topological groups (proposed by Nyikos too) or for topologiacal vector spaces. To my knowledge this question is open for topological vector spaces, and I heard that recently Shibakov proved that there exist topological groups of every countable sequential order, assuming the Continuoum Hypothesis.

Editor's remark:

The author seems to be overstating Shibakov's result. Shibakov constructed, under the Continuum Hypothesis CH, a (Hausdorff) group topology on the group Q of rational numbers which has sequential order $\omega$. Shibakov also asks whether the sequential order of a (Hausdorff) topological group is either $1$ or a limit ordinal less or equal to $\omega_1$. Therefore the result of Pierone for semitopological groups not only remains uncovered by Shibakov's example, but may even turn out to be a specific result for semitopological groups in the future.