Types Of Space Invariance Under The Mapping

It is a main task of General Topology to compare different spaces.Mappings which connect different spaces are important tools to complete it.Which mapping preserves some special generalized metric space is a basic probleme in investigating generalized metric spaces by mappings.G-first countable spaces and g-metri/able spaces have many important topological properities so to investigate which mapping preserves them is very necessary.In [7],Clnian Liu and Mu-ming Dai prove that open-closed mappings preserve g-metri/able spaces;Whether open mappings preserve g-first countable spaces is an open probleme asked by Tanaka in [6].In [4], Sheng-xiang Xia introduces weak opewn mappings and investigates the relations between them and 1-sequence-covering mappings.In the second section of this article ,we investigate weak open mappings have the relations with other mappings and prove that the finite-to-one weak open mappings preserve g-first countable, spaces and weak open closed mapping preserve g-metrizable spaces.In the third section,we investigate an example to show that perfect mappings do not preserve g-first countable spaces , g-metrizable spaces, sn-first countable spaces and sn-metrizable spaces.