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In topological dynamics, the Of course, the statement is preceded by the presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of

It is common knowledge that any topological group that satisfies the lowest separation axiom, T0, is immediately Hausdorff and completely regular; however, this is not the case for normality. This motivates us to introduce the concept of pseudo-normal groups along with pseudo-Tychonoff topological groups as generalizations of the normality and Tychonoffness of topological groups, respectively. We show that every pseudo-normal (resp. pseudo-Tychonoff) topological group is normal (resp. Tychonoff). Generally, the reverse implication of the latter does not hold. Then, we discuss their main properties in detail. To clarify these properties, we provide some examples. Finally, we establish some other results.

Let

In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore different aspects of this connection.

In this paper, we construct uniform structures on a E-compact semilattice of topological groups and study the structure of the uniform completion of a Hausdorff E-compact semilattice of topological groups.

The automorphisms of a two-generator free group \\mathsf F_2 acting on the space of orientation-preserving isometric actions of \\mathsf F_2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group \\Gamma on \\mathbb R ^3 by polynomial automorphisms preserving the cubic polynomial \\kappa _\\Phi (x,y,z) := -x^{2} -y^{2} + z^{2} + x y z -2 and an area form on the level surfaces \\kappa _{\\Phi}^{-1}(k).

In this paper, the notion of a soft topological transformation group is defined and studied. For a soft topological transformation group, it is proven that a map from a soft topological space onto itself is soft homeomorphism. The collection of all soft homeomorphisms of the given soft topological space onto itself constitutes soft topological group under composition. Subsequently, it is proved that there is a homomorphism between soft topological group and the group structure on the collection of all soft homeomorphisms of given topological space. Subsequently, it is shown that the mapping space Map(Y,Y) is soft Hausdorff and verified that any subspace of the mapping space is soft Hausdorff. Additionally, it is proved that the set of all soft homeomorphisms on Y forms a soft discrete space, soft extremally disconnected space, soft Moscow space and a soft Moscow topological group. Later, it is shown that the map from a soft topological group to a mapping space is soft continuous. Finally, it is proved that distinct group structure generates distinct collection of all soft homeomorphisms of the specified soft topological space onto itself is a soft isomorphism.

This volume contains the proceedings of the virtual workshop on Computational Aspects of Discrete Subgroups of Lie Groups, held from June 14 to June 18, 2021, and hosted by the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island.The major theme deals with a novel domain of computational algebra: the design, implementation, and application of algorithms based on matrix representation of groups and their geometric properties. It is centered on computing with discrete subgroups of Lie groups, which impacts many different areas of mathematics such as algebra, geometry, topology, and number theory. The workshop aimed to synergize independent strands in the area of computing with discrete subgroups of Lie groups, to facilitate solution of theoretical problems by means of recent advances in computational algebra.

Let ( U ,  R ) be an approximation space with U being non-empty set and R being an equivalence relation on U , and let G ¯ and G ̲ be the upper approximation and the lower approximation of subset G of U . A topological rough group G is a rough group G = ( G ̲ , G ¯ ) endowed with a topology, which is induced from the upper approximation space G ¯ , such that the product mapping f : G × G → G ¯ and the inverse mapping are continuous. In the class of topological rough groups, the relations of some separation axioms are obtained; some basic properties of the neighborhoods of the rough identity element and topological rough subgroups are investigated. In particular, some examples of topological rough groups are provided to clarify some facts about topological rough groups. Moreover, the version of open mapping theorem in the class of topological rough group is obtained. Further, some interesting open questions are posed.

Let \\(C \\) be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category \\(C \\) is called \\(C \\)-closed if for each morphism \\( X Y\\) in the category \\(C \\) the image \\((X)=\\yın Y: xın X\\;(x,y)ın\\\) is closed in Y. In the paper we survey existing and new results on topological groups, which are \\(C \\)-closed for various categories \\(C \\) of topologized semigroups.