On spaces with point-countable cs-networks

In this paper we discuss three questions about the quotient s-images of metric spaces. The main results are:(1) sequential space with a point-countable cs-network if and only if X is a compact-covering, sequence-covering, quotient and s-image of a metric space. and Y be sequential spaces with point-countable cs-networks, then X × Y is a k-space if and only if one of the three properties below holds.(a) Y are first countable spaces. is a locally compact space. Y are local kw-spaces. X → Y be a pseudo-open s-map. If X is a Fréchet space with a point-countable cs-network, then Y is a Fréchet space with a point-countable cs*-network. artly answer three questions posed by Michael and Nagami (1973), Tanaka (1983), and Gruenhage, Michael and Tanaka (1984) respectively.