\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{Compactification of uniformly continuous mappings} \maketitle \begin{center} Altay A. Borubaev\\ National Academy of Sciences of Kyrgyz Republic, Bishkek, Kyrgyz Republic\\ fiztech-07@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 Below the notion of compactification of a uniformly continuous mapping is introduced and some of their properties are established. The notion of compactification of continuous mappings has been introduced and studied in~\cite{5, 12}. A wider study of compactification of continuous mappings has been done by Pasynkov~\cite{10} and in~\cite{11, 8, 9}. All considered uniform spaces are assumed to be separated and given in coverings terms, mappings are uniformly continuous and topological spaces are Tychonoff. \begin{Definition}~\cite{8}. \rm{Let $f:(X,\mathcal U)\to(Y,\mathcal V)$ be uniformly continuous mapping. A mapping $cf:(cX,c\mathcal U)\to (Y,\mathcal V)$ is called \emph{compactification} or \emph{uniformly perfect extension} of the mapping $f$ if the following conditions hold: 1)$ X\subseteq cX$ ; 2) $[X]_{cX}$; 3)$cf|_{X}=f$; 4) $cf$ is a uniformly perfect mapping}. \end{Definition} For two compactifications $c_{1}f:(c_{1}X,c_{1}\mathcal U)\to (Y,\mathcal V)$ and $c_{2}f:(c_{2}X,c_{2}\mathcal U)\to (Y,\mathcal V)$ of a mapping $f:(X,\mathcal U)\to (Y,\mathcal V)$, as usually we set $c_{2}f\geq c_{1}f$, if there is a uniformly continuous mapping $\varphi:(c_{2}X,c_{2}\mathcal U)\to (c_{1}X,c_{1}\mathcal U)$, such that $c_{2}f=c_{1}f\cdot\varphi$ and $\varphi$ is an identity mapping on $X$. The notions of uniformly perfect and complete mappings are introduced and investigated by the author in~\cite{1,2,3,4}. \begin{Theorem} Every uniformly continuous mapping $f:(X,\mathcal U)\to(Y,\mathcal V)$ has at least one compactification ($\equiv$ of one uniformly perfect extension). \end{Theorem} \begin{Theorem} Every uniformly continuous mapping $f:(X,\mathcal U)\to(Y,\mathcal V)$ has maximal compactification ($\equiv$ a maximum uniformly perfect extension). \end{Theorem} \begin{Theorem} Let $f:(X,\mathcal U)\to(Y,\mathcal V)$ be a uniformly continuous mapping. Then the following conditions are equivalent: \begin{enumerate} \item[{\rm(1)}]A mapping $f$ is uniformly perfect. \item[{\rm(2)}]A mapping $f$ is precompact and for any compact extension $(b_{c}Y,b\mathcal V_{c})$ of a uniform space $(Y,\mathcal V)$ the mapping $b_{c}f$ satisfies to the condition $\beta_{c}f(\beta_{s}X\setminus X)\subseteq b_{c}Y\setminus Y$. \item[{\rm(3)}]A mapping $f$ is precompact and the mapping $\beta_{s}f:(\beta_{s}X,\beta \mathcal U_{s})\to(\beta_{s}Y,\beta\mathcal V_{s})$ satisfies $\beta_{s}f(\beta_{s}X\setminus X)\subseteq\beta_{s}Y\setminus Y$. \item[{\rm(4)}] A mapping $f$ is precompact and there is a compact extension $(b_{c}Y,b\mathcal V_{c})$ of a uniform space $(Y,\mathcal V)$, such that for the extension $\beta_{s}f:(\beta_{s}X,\beta\mathcal U_{s})\to(\beta_{s}Y,\beta\mathcal V_{s})$ of the mapping $f$ the inclusion $\beta_{s}f(\beta_{s}X\setminus X)\subseteq\beta_{s}Y\setminus Y$ holds. \end{enumerate} \end{Theorem} Taking this into account and assuming that $U$ is a maximal precompact uniformity of a Tychonoff space $X$, then Theorem 3 implies well-known theorem of Henriksen and Isbell~\cite{7} in the form, contained in~\cite{6}. The set of all compactifications of a uniformly continuous mapping $f:(X,\mathcal U)\to(Y,\mathcal V)$ will be denoted as $K(f)$. The set $K(f)$ is partially ordered by the order $"\leq"$, which we introduced earlier. A partially ordered set $ (K(f),\leq) $ is not empty (Theorem 1) and has a maximal element (Theorem 2). We denote by $C(f)$ the set of all such uniformities $U_{P}$ of a space $X$ that, firstly $U_{P}\subseteq U$, and, the second, a mapping $f:(X,\mathcal U_{c})\to(Y,\mathcal V)$ is precompact and uniformly continuous. The set $C(f)$ is partially ordered by the inclusion $"\subseteq"$. A partially ordered set $(C(f),\subseteq)$ is not empty and has a maximal element. \begin{Theorem} There is an isomorphism $G:(K(f),\leq)\to (C(f),\subseteq)$ between the partially ordered sets $ (K(f),\leq)$ and $(C(f),\subseteq)$. \end{Theorem} \begin{thebibliography}{39} \bibitem{1}Borubaev A.\,A.,~{\sl Absolutes of uniform spaces}. - Usp. Mat. Nauk, (1988), 43, no. 1, p. 193--194.(in Russian) \bibitem{2}Borubaev A.\,A.,~{\sl Uniformly perfect mappings}. Reports Bolg. Academy of Sciences, (1989), 42, 1, p. 19--23. \bibitem{3}Borubaev A.\,A.,~{\sl Geometry of uniformly continuous maps}. Comment. Academy of Sciences of the GSSR, (1990), 137, 3, p. 497--500. \bibitem{4}Borubaev A.\,A.,~{\sl Uniform topology}. Edited in "Ilim", Bishkek, 2013.(in Russian) \bibitem{5}Cain G.\,L.,~{\sl Compactifications of mappings}. - Proc. Amer. Math. Soc.,(1969), 23, 2, p. 298--303. \bibitem{6}Engelking R.,~{\sl General topology}. Berlin: Heldermann, 1989. 515 p. \bibitem{7}Henriksen M.,~Isbell J.\,R.,~{\sl Some properties of compactifications}. - Duke Math. J., (1958), 25, p. 83--106. \bibitem{8}Ormotsadze R.\,N.,~{\sl On points of closedness of mapping}. - Comment. Academy of Sciences of the GSSR, 135, 2, p. 277--280.(in Russian) \bibitem{9}Ormotsadze R.\,N.,~{\sl On perfect maps}. - Comment. Academy of Sciences of the GSSR, (1985), 119, 1, p. 25--28.(in Russian) \bibitem{10}Pasynkov B.\,A.,~{\sl On extending onto mappings some concepts and statements concerning spaces}. In the collection "Mappings and functors". MSU,(1984), p. 72--102. (in Russian) \bibitem{11}Ulyanov V.\,M.,~{\sl On compactifications of countable character and absolutes}.- Matem. Sb.,(1975),98, 2, p. 223--254.(in Russian) \bibitem{12}Whyburn G.\,T.,~{\sl A unified space of mappings}. - Trans. Amer. Soc., (1953), 74, p. 344--350. \end{thebibliography} \end{document}