Previous |  Up |  Next


real-valued function; Stone-Weierstrass; uniform approximation; Lindelöf space; locally in
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 $\Delta$.
[BH74] Blair R., Hager A.: Extension of zero-sets and real-valued functions. Math. Z. 136 (1974), 41–52. DOI 10.1007/BF01189255 | MR 0385793
[D95] Darnel M.: Theory of Lattice-ordered Groups. Marcel Dekker, New York, 1995. MR 1304052 | Zbl 0810.06016
[E89] Engelking R.: General Topology. Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[FGL65] Fine N., Gillman L., Lambek J.: Rings of quotients of rings of functions. McGill Univ. Press, 1965; republished by Network RAAG, 2005. MR 0200747 | Zbl 0143.35704
[GJ60] Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand, Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199 | Zbl 0327.46040
[H69] Hager A.: On inverse-closed subalgebras of $C(X)$. Proc. London Math. Soc. 19 (1969), 233–257. DOI 10.1112/plms/s3-19.2.233 | MR 0244948 | Zbl 0169.54005
[H76] Hager A.: A class of function algebras (and compactifications, and uniform spaces). Sympos. Math. 17 (1976), 11–23. MR 0425891 | Zbl 0353.46014
[H$\infty$] Hager A.: *-maximum $l$-groups. in preparation.
[HM02] Hager A., Martinez J.: $C$-epic compactifications. Topology Appl. 117 (2002), 113–138. DOI 10.1016/S0166-8641(00)00119-X | MR 1875905 | Zbl 0993.54024
[HR77] Hager A., Robertson L.: Representing and ringifying a Riesz space. Sympos. Math. 21 (1977), 411–431. MR 0482728 | Zbl 0382.06018
[HR78] Hager A., Robertson L.: Extremal units in an Archimedean Reisz space. Rend. Sem. Mat. Univ. Padova 59 (1978), 97–115. MR 0547081
[HJ61] Henriksen M., Johnson D.: On the structure of a class of lattice-ordered algebras. Fund. Math. 50 (1961), 73–94. MR 0133698
[H47] Hewitt E.: Certain generalizations of the Weierstrass Approximation Theorem. Duke Math. J. 14 (1947), 419–427. MR 0021662 | Zbl 0029.30302
[N73] Nyikos P.: Prabir Roy's space $\Delta$ is not $\mathbb N$-compact. General Topology and Appl. 3 (1973), 197–210. DOI 10.1016/0016-660X(72)90012-8 | MR 0324657
[R68] Roy P.: Nonequality of dimensions for metric spaces. Trans. Amer. Math. Soc. 134 (1968), 117–132. DOI 10.1090/S0002-9947-1968-0227960-2 | MR 0227960 | Zbl 0181.26002
[S87] Sola M.: Roy's space $\Delta$ and its $\mathbb N$-compactification. Thesis, Univ. of S. Carolina, 1987.
[S48] Stone M.: The generalized Weierstrass approximation theorem. Math. Mag. 21 (1948), 167–184. DOI 10.2307/3029750 | MR 0027121
Partner of
EuDML logo