When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be robust dividing line in complexity topological dynamics $G$. We introduce class groups, CAP which provides neat generalization this all groups. prove number characterizations class, having very different flavors, and use these that groups enjoys nice closure properties. As concrete application, we compu...

‎‎The purpose of this paper is to introduce new methods for constructing generalized groups, generalized topological groups and top spaces. We study some properties of these structures and present some relative concrete examples. Moreover, we obtain generalized groups by using of Hilbert spaces and tangent spaces of Lie groups, separately.

We introduce a natural generalization of Borel’s Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCא0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture,¬BCא1 is equivalent to the existence of a Kurepa tree of heightא1. Using the connection of BCκ with a generalization of Kurepa’s Hypothesis, we obtain the following...

We investigate some properties of topological groups related to disconnectedness or Archimedeanness. We prove or disprove the preservation of those under operations as subgroups, quotients, products, etc. Characterizations of non-Archimedeanness are obtained by using embedding into universal groups. We also clarify the differences between the properties by constructing miscellaneous Polish groups.

The terminology used in the statement of this theorem, and in what follows, is that of our two previous papers on generalized manifolds [l, 2] , and we assume that the reader is familiar with them. We make, however, one change. We find it convenient to define infinite cycles in the following way: We add to G an ideal point, g, taking as neighborhoods of g those open subsets of G whose closures ...