The topological structure of (homogeneous) spaces and groups with countable cs∗-character

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 N-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 kω- subgroup. This result is specific for topological groups: under Martin Axiom there exists a sequential topologically homogeneous kω-space X with N0 = cs∗x (X) <ψ (X)