On sequential analytic groups

Abstract

We answer a question of S.~Todorčević and C.~Uzcátegui from \cite{TU1} by showing that the only possible sequential orders of sequential analytic groups are $1$ and ${\omega }_1$. Other results on the structure of sequential analytic spaces and their relation to other classes of spaces are given as well. In particular, we provide a full topological classification of sequential analytic groups by showing that all such groups are either metrizable or $k_{\omega }$-spaces, which, together with a result by Zelenyuk, implies that there are exactly ${\omega }_1$ non homeomorphic analytic sequential group topologies.