A Study on Topological Structures of Transformation Groups

Abstract

The topological transformation groups, where continuity conditions in topological newlinegroups and in action map are replaced with irresoluteness is studied in this thesis. It begins newlineby examining irresolute functions and their interaction with separation axioms. The Irrtopological newlinetransformation groups, Irr and#8727; -topological transformation groups, I and#8727; -topological newlinetransformation groups and I-topological transformation groups are introduced. The study newlinealso explores the set of all irresolute functions forms a topological group. newlineThe isotropy groups and normalizers in I-topological transformation groups are newlinestudied. Additionally, the study introduces s-convergence and examines its properties newlinealong with semi-closed sets. The fixed point set and equivariant maps of I-topological newlinetransformation groups are investigated, proving that the fixed point set is semi-open. The newlineorbit spaces of these groups are analyzed and an H-semi-homeomorphism is established. newlineIt also explores the homogeneous spaces of I-topological transformation groups, newlineproving that the natural projection from a quotient group to its homogeneous space is newlineboth an H-isomorphism and irresolute. An induced I-transformation group is constructed newlinethrough an induced K-action, its relationships with isotropy groups and orbit spaces are newlineanalyzed newline