In this paper we introduce and study three new cardinal topological
invariants called the cs*, cs-, and sb-characters. The class of topological
spaces with countable cs*-character is closed under many topological operations
and contains all aleph-spaces and all spaces with point-countable cs*-network.
Our principal result states that each non-metrizable sequential topological
group with countable cs*-character has countable pseudo-character and contains
an open $k_\omega$-subgroup.