The Theory of Topo-Symmetric Extensions of Topological Groups

We introduce the notion of \emph{topo-symmetric extensions} of topological groups, a new generalization of classical group extensions that incorporates both topological and symmetry constraints. We define morphisms between such extensions, construct the associated groupoid, and develop classification results in terms of adapted cohomology. Several new invariants are introduced, including dimension, stabilizer, and density invariants, which characterize the fine structure of these extensions. Applications are given for finite groups, compact Lie groups, and profinite groups. This theory extends classical cohomological correspondence theorems \cite{MacLane1963, EilenbergMacLane1947}, while opening new perspectives in arithmetic, asymptotic distribution, and congruence properties \cite{HardyWright2008, Serre1979}. Finally, we propose conjectures and open problems concerning density, maximal orders, and modular distribution of topo-symmetric extensions.