On sequential analytic groups

TL;DR

This paper classifies sequential analytic groups, showing they can only have sequential orders 1 or , and that all such groups are either metrizable or $k_$-spaces, with exactly  non-homeomorphic topologies.

Contribution

It provides a complete topological classification of sequential analytic groups, resolving an open question about their possible sequential orders.

Findings

Sequential orders of analytic groups are only 1 or .

All sequential analytic groups are either metrizable or $k_$-spaces.

There are exactly  non-homeomorphic analytic sequential group topologies.

Abstract

We answer a question of S.~Todor\v{c}evi\'c and C.~Uzc\'ategui 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.