Metrizability of sequential topological groups with point-countable 𝑘-networks

Abstract

Abstract. We prove that a Hausdorff sequential topological group with a point-countable k-network is metrizable iff its sequential order is less than ω 1 . In the non Hausdorff case metrizability may be replaced by σ-locally finite base.

On sequential analytic groups

“…It answers Question 7.1 from [15]. Theorem 1 below together with a result in [13] can also be used to obtain this statement.…”

“…Even in the case of a countable X, not every ℵ 0 -space is a k ωspace, however, when X is a countable sequential non Fréchet topological group, a corollary of a more general result in [1] implies that X is a k ω -space if and only if X is an ℵ 0 -space. A result in [13] shows that for each such group so(X) = ω 1 . Perhaps the most surprising property of the class of all k ω countable group topologies is that there are exactly ω 1 of them, moreover, the topological type of such group is uniquely described by the supremum of Cantor-Bendixson ranks of its compact subspaces (see [17] and [5]).…”

On large sequential groups

“…Thus one can assume that all r n 's are integers, or linearly independent over Q. It is easy to see that in such a topology R becomes a sequential non Fréchet group (indeed, its sequential order is ω 1 , see [14]). It is also an easy observation that any countable closed subgroup of R must be cyclic and that every cyclic subgroup of R is dense in itself in the new topology.…”

A note on rectifiable spaces

“…Lemma 6.2. [23] A sequential non-Fréchet-Urysohn space with a point-countable k-network contains a closed copy of S 2 . Lemma 6.3.…”

“…Lemma 6.5. [23] Let X be a sequential space with a point-countable k-network such that for any x ∈ X and U ⊂ X the property P (x, U ) holds. Then for any α < ω 1 , x ∈ X, U ⊂ X open in X the following property holds:…”