Abstract: By the ponomarev's way and the definit ion of msss- mapping, this paper studies the point- countable covering problem on the generalized metric space, proving that the spaces with σ-locally countable cs^* - network and csnetwork can be characterized by the image of metrizable spaces under sequence- covering mappings and strong sequence-covering mappings.Thus, the study of the spaces with σ-locally countable collect ionwise is made more complete.

Key words: sequence-covering mapping, strong sequence- covering mapping, msss- mapping, cs- net , cs^*- net