Abstract: In this paper the relation between the σ-images of metrical spaces and spaces with σ-locally finite cs-network, or spaces with σ-locally finite cs*-network, or spaces with σ-locally finite sequence neighborhood network, or spaces with σ-locally finite sequence open network are established by use of σ-mapping.

Key words: σ-mapping, o-locally finite family of sets, cs-network, cs"-networks, sequence neighborhood network, sequence open network