On large sequential groups

2016/04/29 by Shibakov, Alexander
#22A05 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1604.08874

Abstract

We construct, using \diamondsuit, an example of a sequential group G such that the only countable sequential subgroups of G are closed and discrete, and the only quotients of G that have a countable pseudocharacter are countable and Fréchet. We also show how to construct such a G with several additional properties (such as make G² sequential, and arrange for every sequential subgroup of G to be closed and contain a nonmetrizable compact subspace, etc.). Several results about k_ω sequential groups are proved. In particular, we show that each such group is either locally compact and metrizable or contains a closed copy of the sequential fan. It is also proved that a dense proper subgroup of a non Fréchet k_ω sequential group is not sequential extending a similar observation of T.~Banakh about countable k_ω groups.

Related

• Countably compact groups and sequential order
• No interesting sequential groups
• Precompact groups and convergence
• Real-valued measurable cardinals and sequentially continuous homomorphisms
• Convergence in topological groups and the Cohen reals