Topological \(\text{AE}(0)\)-groups (Q2717643)

The author studies AE(0)-groups in the hope that his results indicate a potential for a non-trivial theory that unifies and generalizes the theories of locally compact and Polish groups. The author gives a spectral characterization of AE(0)-groups in terms of well-ordered continuous inverse spectra. Also, he characterizes 0-soft homomorphisms with Polish kernels between AE(0)-groups. Then he proves the existence of universal AE(0)-groups of a given weight and the existence of universal actions of AE(0)-groups of a given weight on compact AE(0)-spaces of the same weight. The author characterizes AE(0)-groups that are isomorphic to closed subgroups of powers \(S^r_\infty\) of the symmetric groups of all bijections of \(\mathbb{N}\) under the relative topology inherited from \(\mathbb{N}^\mathbb{N}\). Finally, he proves that every AE(0)-group is Baire isomorphic to a product of Polish groups.