# Article

 Title: Local/global uniform approximation of real-valued continuous functions (English) Author: Hager, Anthony W. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 52 Issue: 2 Year: 2011 Pages: 283-291 Summary lang: English . Category: math . Summary: For a Tychonoff space $X$, $C(X)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^{*}(X)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^{*}(X)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C(X)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent to: Every compact space has AP. It is shown here that the class of spaces with AP contains all Lindelöf spaces and is closed under formation of topological sums. Thus, any locally compact paracompact space has AP. A paracompact space failing AP is Roy's completely metrizable space $\Delta$. (English) Keyword: real-valued function Keyword: Stone-Weierstrass Keyword: uniform approximation Keyword: Lindelöf space Keyword: locally in MSC: 06F20 MSC: 26E99 MSC: 41A30 MSC: 46E05 MSC: 54C30 MSC: 54C35 MSC: 54D20 MSC: 54D35 idZBL: Zbl 1240.54062 idMR: MR2849050 . Date available: 2011-05-17T08:41:52Z Last updated: 2013-09-22 Stable URL: http://hdl.handle.net/10338.dmlcz/141501 . Reference: [BH74] Blair R., Hager A.: Extension of zero-sets and real-valued functions.Math. Z. 136 (1974), 41–52. MR 0385793, 10.1007/BF01189255 Reference: [D95] Darnel M.: Theory of Lattice-ordered Groups.Marcel Dekker, New York, 1995. Zbl 0810.06016, MR 1304052 Reference: [E89] Engelking R.: General Topology.Heldermann, Berlin, 1989. Zbl 0684.54001, MR 1039321 Reference: [FGL65] Fine N., Gillman L., Lambek J.: Rings of quotients of rings of functions.McGill Univ. Press, 1965; republished by Network RAAG, 2005. Zbl 0143.35704, MR 0200747 Reference: [GJ60] Gillman L., Jerison M.: Rings of Continuous Functions.Van Nostrand, Princeton, N.J.-Toronto-London-New York, 1960. Zbl 0327.46040, MR 0116199 Reference: [H69] Hager A.: On inverse-closed subalgebras of $C(X)$.Proc. London Math. Soc. 19 (1969), 233–257. Zbl 0169.54005, MR 0244948, 10.1112/plms/s3-19.2.233 Reference: [H76] Hager A.: A class of function algebras (and compactifications, and uniform spaces).Sympos. Math. 17 (1976), 11–23. Zbl 0353.46014, MR 0425891 Reference: [H$\infty$] Hager A.: *-maximum $l$-groups.in preparation. Reference: [HM02] Hager A., Martinez J.: $C$-epic compactifications.Topology Appl. 117 (2002), 113–138. Zbl 0993.54024, MR 1875905, 10.1016/S0166-8641(00)00119-X Reference: [HR77] Hager A., Robertson L.: Representing and ringifying a Riesz space.Sympos. Math. 21 (1977), 411–431. Zbl 0382.06018, MR 0482728 Reference: [HR78] Hager A., Robertson L.: Extremal units in an Archimedean Reisz space.Rend. Sem. Mat. Univ. Padova 59 (1978), 97–115. MR 0547081 Reference: [HJ61] Henriksen M., Johnson D.: On the structure of a class of lattice-ordered algebras.Fund. Math. 50 (1961), 73–94. MR 0133698 Reference: [H47] Hewitt E.: Certain generalizations of the Weierstrass Approximation Theorem.Duke Math. J. 14 (1947), 419–427. Zbl 0029.30302, MR 0021662 Reference: [N73] Nyikos P.: Prabir Roy's space $\Delta$ is not $\mathbb N$-compact.General Topology and Appl. 3 (1973), 197–210. MR 0324657, 10.1016/0016-660X(72)90012-8 Reference: [R68] Roy P.: Nonequality of dimensions for metric spaces.Trans. Amer. Math. Soc. 134 (1968), 117–132. Zbl 0181.26002, MR 0227960, 10.1090/S0002-9947-1968-0227960-2 Reference: [S87] Sola M.: Roy's space $\Delta$ and its $\mathbb N$-compactification.Thesis, Univ. of S. Carolina, 1987. Reference: [S48] Stone M.: The generalized Weierstrass approximation theorem.Math. Mag. 21 (1948), 167–184. MR 0027121, 10.2307/3029750 .

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