Topological groups with a certain point-countable cover

Shibakov proved in 1998 that every sequential topological group with a point-countable k-network is metrizable if its sequential order is less than ω1. In this paper, we study further topological properties of sequential topological groups with a point-countable k-network. We introduce the notion of a cs-cover, and show that: If G is a sequential topological group with a point-countable cs-cover F such that any product of finitely many elements of F is Lindelöf, then G has an open subgroup which is the union of countably many elements of F. In addition, we show that every topological group with a σ-hereditarily closure-preserving k-network is an ℵ-space.