Searcharxiv⌕ Search

arXiv subjects

K. L. Kozlov

Publications and source records attributed to K. L. Kozlov.

6 recordsLinked to original sources
arXiv2026

On proper compactifications of topological groups

In the present paper, we examine in detail the method of "graph compactifications" of topological groups. The graph and Ellis methods of constructing proper compactifications of topological groups are applied for the investigation of possible extensions of algebraic operations on a topological group to its compactifications, and give descriptions of Roelcke, Ellis, WAP, and graph compactifications of topological groups. Additionally, using dichotomy theorems of A.V.Arhangelskii, we show that the description of compactifications can be effectively used in the investigation of topological properties of their remainders. As examples, subgroups of the permutation group (in the permutation topology) and the automorphism group of a LOTS (in the topology of pointwise convergence) are examined.

math.GN↗
arXiv2025

Enveloping semigroups as compactifications of topological groups

Ellis's "functional approach" allows one to obtain proper compactifications of a topological group $G$ if $G$ can be represented as a subgroup of the homeomorphism group of a space $X$ in the topology of pointwise convergence and $G$-space $X$ is $G$-Tychonoff. These compactifications, called Ellis compactifications, are right topological monoids and $G$-compactifications of the group $G$ with its action by multiplication on the left on itself. A comparison is made between Ellis compactifications of $G$ and the Roelcke compactification of $G$. Uniformity corresponding to the Ellis compactification of $G$ for its representation in a compact space $X$ is established. Proper Ellis semigroup compactifications are described for groups ${\rm S}(X)$ (the permutation group of a discrete space $X$) and ${\rm Aut} (X)$ (automorphism group of an ultrahomogeneous chain $X$) in the permutation topology and ${\rm Aut} (X)$ of LOTS $X$ in the topology of pointwise convergence.

math.GN↗
arXiv2025

Enveloping Ellis semigroups as compactifications of transformations groups

The notion of a proper Ellis semigroup compactification is introduced. Ellis's functional approach shows how to obtain them from totally bounded equiuniformities on a phase space $X$ when the acting group $G$ is with the topology of pointwise convergence and the $G$-space $(G, X, \curvearrowright)$ is $G$-Tychonoff. The correspondence between proper Ellis semigroup compactifications of a topological group and special totally bounded equiuniformities (called Ellis equiuniformities) on a topological group is established. The Ellis equiuniformity on a topological transformation group $G$ from the maximal equiuniformity on a phase space $G/H$ in the case of its uniformly equicontinuous action is compared with Roelcke uniformity on $G$. Proper Ellis semigroup compactifications are described for groups $S\,(X)$ (the permutation group of a discrete space $X$) and $Aut\,(X)$ (automorphism group of an ultrahomogeneous chain $X$) in the permutation topology. It is shown that this approach can be applied to the unitary group of a Hilbert space.

math.GN↗
arXiv2025

On two methods of constructing compactifications of topological groups

The classification of (proper) compactifications of topological groups with respect to the possibility of extensions of algebraic operations is presented. Ellis' method of construction compactifications of topological groups allows one to obtain all right topological semigroup compactifications on which the multiplication on the left continuously extends. Presentation of group elements as graphs of maps in the hyperspace with Vietoris topology allows one to obtain compactifications on which the involution and the multiplication on the left extend.

math.GN↗
arXiv2002

The maximal G-compactifications of G-spaces with special actions

An action on a G-space induces uniformities on the phase space. It is shown when the maximal G-compactification of a G-space can be obtained as a completion of the phase space with respect to one of these uniformities. Structure of G-spaces with special actions is investigated.

math.GN↗