Homogeneous Spaces and Induced Transformation Groups of S-Topological Transformation Group

This paper explores the homogeneous spaces and induced transformation groups of S-topological transformation group.S-topological transformation group is a structure constructed by concatenating a topological group with a topological space through a semi totally continuous action.It is shown that any map from a topological group to the quotient group of a finite H ausdorff t opological g roup b y t he isotropy group is surjective, continuous, open and it has been proven that any map from the quotient group of a finite Hausdorff topological group by the isotropy group to the homogenous space is both H-isomorphism and semi totally continuous.Furthermore, an equivariant map has been established between homogeneous spaces and discussed the partial order relation on the family of all Hausdorff homogeneous spaces for a compact Hausdorff topological group.Subsequently, an induced S-topological transformation group is constructed by an induced H-action.For any compact subgroup K of a topological group H, it is verified t hat a ny m ap from the topological spcae Y to the orbit space of K-action is continuous and a K-map.For any H-space, K-map and an induced S-topological transformation group; it is proved that there is a unique semi totally continuous H-map.Additionally, it is shown that for a topological group, a subgroup K of topological group and a K-space, there is a unique H-space and a unique injective K-map and also it is established that for a H-space and a semi totally continuous K-map, there exists a unique semi totally continuous H-map.Finally, it is demonstrated that for a finite H ausdorff topological group, finite Frechet space and a M -space, any map from the orbit space of M -action to H × N (N × M Y ) is semi totally continuous, for the subgroups M and N of topological group.