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Имя: Чекеев Асылбек Асакеевич Название: On $\beta-$like compactifications and inversion-closed rings of uniform spaces
Город, страна: Бишкек, Кыргызстан
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\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}
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\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}
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