Previous |  Up |  Next

Article

Keywords:
frame of reals; lattice ordered rings of real valued continuous functions and integer valued continuous functions; extremally disconnected frame; basically disconnected frame; cozero map
Summary:
This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or $\sigma$-comp\-lete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.
References:
[1] Ball R.N., Walters-Wayland J.L.: $C$- and $C^*$-embedded sublocales. Dissertationes Math. 412 (2002). MR 1952051 | Zbl 1012.54025
[2] Banaschewski B.: Universal zero-dimensional compactifications. In Categorical Topology and its Relation to Analysis, Algebra, and Combinatorics, Prague, Czechoslovakia, August 1988, World Scientific, Singapore, 1989, pp. 257-269. MR 1047906
[3] Banaschewski B.: The frame envelope of a $\sigma$-frame. Quaest. Math. 16 (1993), 51-60. MR 1217474 | Zbl 0779.06009
[4] Banaschewski B.: The real numbers in pointfree topology. Textos de Matemática Série B, No. 12, Departamento de Matemática da Universidade de Coimbra, 1997. MR 1621835 | Zbl 0891.54009
[5] Banaschewski B., Mulvey C.J.: Stone-Čech compactification of locales II. J. Pure Appl. Algebra 33 (1984), 107-122. MR 0754950 | Zbl 0549.54017
[6] Gillman L., Jerison M.: Rings of continuous functions. D. Van Nostrand, 1960. MR 0116199 | Zbl 0327.46040
[7] Johnstone P.T.: Conditions related to De Morgan's law. In Applications of Sheaves, Proceedings, Durham, 1977, Springer LNM 753 (1979), pp. 479-491. MR 0555556 | Zbl 0445.03041
[8] Johnstone P.T.: Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982. MR 0698074
[9] Nakano H.: Über das System aller stetigen Funktionen auf einen topologischen Raum. Proc. Imp. Acad. Tokyo 17 (1941), 308-310. MR 0014173
[10] Stone M.H.: Boundedness properties in function lattices. Canad. J. Math. 1 (1949), 176-186. MR 0029091 | Zbl 0032.16901
[11] Vickers S.: Topology via Logic. Cambridge Tracts in Theor. Comp. Sci. No 5, Cambridge University Press, 1985. MR 1002193 | Zbl 0922.54002
Partner of
EuDML logo