Abstract: In this paper, spaces with a locally countable sn-network are discussed. It is shown that a space with a locally countable sn-network iff it is an snf-countable space with a locally countable k-network. As its application, almost-open and closed mappings(or finite-to-one and closed mapping) preserve locally countable sn-networks, and a perfect preimage theorem on spaces with a locally countable sn-network is established.

Key words: locally countable family, sn-network, k-network, cs-network