Pointfree topology version of image of real-valued continuous functions

Document Type: Research Paper


1 Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

3 Faculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.


Let $ { \mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree  version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree  version of $C_c(X).$
The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in $ {\mathcal{R}} L$. In particular, we will introduce the pointfree version of the ring $C_c(X)$. We define a relation from $ {\mathcal{R}} L$ into the power set of $\mathbb R$, namely overlap. Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as $\mathop{\hbox{Im}} (f) \subseteq S$ for every continuous function $f:X\rightarrow\mathbb R$ and $ S \subseteq \mathbb R$.


1] Ball, R.N. and Hager, A.W., On the localic Yoshida representation of an archimedean lattice ordered group with weak order unit, J. Pure Appl. Algebra, 70 (1991), 17-43.
[2] Ball, R.N. and Walters-Wayland, J., C- and C^* quotients on pointfree topology, Dissertations Mathematicae (Rozprawy Mat), 412 Warszawa (2002), 62 pp.
[3] Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered algebraic structures, (Curacao 1995), Kluwer Acad. Publ. (1997), 123-148.
[4] Banaschewski, B., The real numbers in pointfree topology, Textos Mat. Sér. B 12, University of Coimbra, 1997.
[5] Banaschewski, B. and Gilmour, C.R.A., Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 577-587.
[6] Bhattacharjee, P., Knox, M.L., and McGovern,W.W., The classical ring of quotients of Cc(X), Appl. Gen. Topol. 15(2) (2014), 147-154.
[7] Dube, T. and Ighedo, O., On z-ideals of pointfree function rings, Bull. Iran. Math. Soc. 40 (2014), 657-675.
[8] Dube, T., Iliadis, S., Van Mill, J., and Naidoo, I., A Pseudocompact completely regular frame which is not spatial, Order 31(1) (2014), 115-120.
[9] Ebrahimi, M.M. and Karimi Feizabadi, A., Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291-299.
[10] Ebrahimi, M.M. and Mahmoudi, M., “Frames”, Technical Report, Department of Mathematics, Shahid Beheshti University, 1996.
[11] Estaji, A.A., Karimi Feizabadi, A., and Abedi, M., Zero sets in pointfree topology and strongly z-ideals, Bull. Iran. Math. Soc 41(5) (2015), 1071-1084.
[12] Estaji, A.A., Karimi Feizabadi, A., and Robat Sarpoushi, M., zc-Ideals and prime ideals in the ring RcL, to appear in Filomat.
[13] Ghadermazi, M., Karamzadeh, O.A.S., and Namdari, M., On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova 129 (2013), 47-69.
[14] Gillman, L. and Jerison, M., “Rings of continuous functions”, Springer-Verlag, 1976.
[15] Johnstone, P.T., “Stone spaces”, Cambridge University Press, 1982.
[16] Karamzadeh, O.A.S., Namdari, M., and Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol. 16 (2015), 183-207.
[17] Karamzadeh, O.A.S. and Rostami, M., On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184.
[18] Namdari, M. and Veisi, A., Rings of quotients of the subalgebra of C(X) consisting of functions with countable image, Inter. Math. Forum 7 (2012), 561-571.
[19] Picado, J. and Pultr, A., “Frames and Locales: topology without points”, Birkhäuser/Springer, Basel AG, 2012.