Александровские чтения-2016 | 22-26 мая 2016
Участник: Чекеев Асылбек Асакеевич
Имя: Чекеев Асылбек Асакеевич
Название: On $\beta-$like compactifications and inversion-closed rings of uniform spaces
Город, страна: Бишкек, Кыргызстан
\documentclass[12pt,a4paper,reqno]{amsart} \renewcommand{\baselinestretch}{1.2} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{anysize} \marginsize{2.cm}{2.cm}{1.cm}{1.cm} \begin{document} \title{On $\beta-$like compactifications and inversion-closed rings\\ of uniform spaces} \maketitle \begin{center} Asylbek A. Chekeev\\ Kyrgyz National University, Mathematics, Informatics and Cybernetics Faculty, Kyrgyz-Turkish Manas University, Faculty of Science, Bishkek, Kyrgyz Republic\\ asyl.ch.top@mail.ru \end{center} %REFERENCES ARE NECESSARY IN THE THESIS! %NO GRAPHICS, FIGURES! \newtheorem{Theorem}{Theorem} \newtheorem{Prop}{Proposition} \newtheorem{Definition}{Definition} \newtheorem{Lemma}{Lemma} \newtheorem{Proof}{Proof} \newtheorem{Hypothesis}{Hypothesis} \bigskip In~\cite{3} for uniform space $uX$ by a normal base ${\mathcal Z}=\{\textbf{Z}(f)=f^{-1}(0): f \in U(uX)\}$, where $U(uX)$ is a set of all uniformly continuous functions on $uX$ Wallman compactification $\beta_{u}X$~\cite{1, 8} and Wallman realcompactification $\upsilon_{u}X$~\cite{11} had been constructed and their uniformities had been described. The compactification $\beta_{u}X$ is a $\beta-$like compactification~\cite{10}, and it is connected with an algebra $C_{u}(X) (C^{*}_{u}(X))$ of all (bounded) $u-$continuous functions on $uX$ in sense~\cite{2, 4}. We note, that $C_{u}(X)$ is an algebra with inversion in sense~\cite{6, 7, 9}. Now for the rings $C_{u}(X)$ and $C^{*}_{u}(X)$ a uniform analogues of Gelfand--Kolmogoroff and Stone Theorems have been proved, which are established a homeomorphism between $\beta_{u}X$ and a space of maximal ideals of the rings $C_{u}(X)$ and $C^{*}_{u}(X)$ with Stone topology. Due to Wallman realcompactification $\upsilon_{u}X$ $z_{u}-$complete uniform spaces are determined and a uniform analogue of Hewitt Theorem~\cite[3.12.21(g)]{5} have been proved. \begin{thebibliography}{99} \bibitem{1}Aarts J.\,M.,~Nishiura T.,~{\sl Dimension and Extensions},- North-Holland, 1993. 331 p. \bibitem{2}Charalambous M.\,G.,~{\sl A new covering dimension function for uniform spaces}, J. London Math. Soc. 2(11) (1975), p. 137--143. \bibitem{3}Chekeev A.\,A.,~{\sl Uniformities for Wallman compactifications and realcompactifications},- Topology Appl., V.201., (2016), p.145--156 \bibitem{4}Frolik Z.,~{\sl A note on metric-fine spaces}, Proc. Amer. Math.Soc.,V. 46, n.1, (1974), p.111--119. \bibitem{5}Engelking R.,~{\sl General Topology}, Berlin: Heldermann, 1989. 515 p. \bibitem{6}Hager A.\,W.,~Johnson D.\,J.,~{\sl A note on certain subalgebras of $C(X)$}, Canad. J. Math. 20(1968), p. 389--393. \bibitem{7}Hager A.W.,~{\sl On inverse-closed subalgebra of C(X)}, Proc. Lond. Math. Soc. 19(3)(1969), p. 233--257. \bibitem{8}Iliadis S.\,D.,~{\sl Universal spaces and mappings}, North-Holland Mathematics Studies, 198. Elsevier Science B.V., 2005, Amsterdam. 559 p. \bibitem{9}Isbell J .\,R.,~{\sl Algebras of uniformly continuous functions}, Ann. of Math., 68 (1958), p. 96--125. \bibitem{10} Mr\'owka S.,~{\sl $\beta-$like compactifications}, Acta Math. Acad. Sci. Hungaricae, 24 (3-4)(1973), p. 279--287. \bibitem{11} Steiner A.\,K.,~Steiner E.\,F.,~{\sl Nest generated intersection rings in Tychonoff spaces}, Trans. Amer. Math. Soc. 148 (1970), p. 589--601. \end{thebibliography} \end{document}

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