Some Results On Spaces With Sn-Network

This paper is composed of three parts. Firstly, the sn-open mapping is introduced. By using it, certain sn-first countable spaces are characterized as images of metric spaces under various sn-open mappings. Also, this part give the relationship between sn-open mapping and 1-sequence-covering mapping. Secondly, The characterization of 1-sequence covering cs-Ï€ images of metric spaces and that of compact-covering cs-Ï€ images of metric spaces are established. By using those, a characterization to development spaces with point-countable (or compact-countable) bases is given. Finally, spaces with a locally countable sn-network are discussed. It is shown that a space with a locally countable sn-network if and only if is an snf-countable space with a locally countable k-network. By using the result, finite-to-one and closed mappings preserve spaces with locally countable sn-network is proved. And a perfect preimage theorem on spaces with a locally countable sn-network is established.