In this paper, we discuss a class of spaces with a locally countable
sn-network. We give some characterizations of this class and establish the
relation among spaces with a locally countable weak-base, spaces with a locally
countable sn-network and spaces with a locally countable cs-network. Also, we
investigate variance and inverse invariance of spaces with a locally countable
sn-network under certain mappings. As some applications of these results, we
obtain some results relative to spaces with a locally countable weak-base.