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Locally functionally countable subalgebra of $$\mathcal{R}(L)$$. (English) Zbl 07250674
Summary: Let $$L_c(X)=\lbrace f\in C(X)\colon\overline{C_f}=X\rbrace$$, where $$C_f$$ is the union of all open subsets $$U\subseteq X$$ such that $$\vert f(U)\vert\le\aleph_0$$. In this paper, we present a pointfree topology version of $$L_c(X)$$, named $$\mathcal{R}_{\ell c}(L)$$. We observe that $$\mathcal{R}_{\ell c}(L)$$ enjoys most of the important properties shared by $$\mathcal{R}(L)$$ and $$\mathcal{R}_c(L)$$, where $$\mathcal{R}_c(L)$$ is the pointfree version of all continuous functions of $$C(X)$$ with countable image. The interrelation between $$\mathcal{R}(L)$$, $$\mathcal{R}_{\ell c}(L)$$, and $$\mathcal{R}_c(L)$$ is examined. We show that $$L_c(X)\cong\mathcal{R}_{\ell c}\big (\mathfrak{O}(X)\big)$$ for any space $$X$$. Frames $$L$$ for which $$\mathcal{R}_{\ell c}(L)=\mathcal{R}(L)$$ are characterized.
##### MSC:
 06D22 Frames, locales 54C05 Continuous maps 54C30 Real-valued functions in general topology
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##### References:
 [1] Azarpanah, F.; Karamzadeh, O. A.S.; Keshtkar, Z.; Olfati, A. R., On maximal ideals of $$C_c(X)$$ and the uniformity of its localizations, Rocky Mountain J. Math. 48 (2) (2018), 354-384, http://doi.org/10.1216/RMJ-2018-48-2-345 [2] Ball, R. N.; Walters-Wayland, J., $$\text{C}$$- and $$\text{C}^*$$- quotients in pointfree topology, Dissertationes Math. (Rozprawy Mat.) 412 (2002), 354-384 [3] Banaschewski, B., The real numbers in pointfree topology, Textos de Mathemática (Séries B), Universidade de Coimbra, Departamento de Mathemática, Coimbra 12 (1997), 1-96 [4] Bhattacharjee, P.; Knox, M. L.; Mcgovern, W. W., The classical ring of quotients of $$C_c(X)$$, Appl. Gen. Topol. 15 (2) (2014), 147-154, https://doi.org/10.4995/agt.2014.3181 [5] Dowker, C. H., On Urysohn’s lemma, General Topology and its Relations to Modern Analysis, Proceedings of the second Prague topological symposium, 1966, Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 1967, pp. 111-114 [6] Estaji, A. A.; Karimi Feizabadi, A.; Robat Sarpoushi, M., $$z_c$$-ideals and prime ideals in the ring $$\mathcal{R}_c L$$, Filomat 32 (19) (2018), 6741-6752, https://doi.org/10.2298/FIL1819741E [7] Estaji, A. A.; Karimi Feizabadi, A.; Zarghani, M., Zero elements and $$z$$-ideals in modified pointfree topology, Bull. Iranian Math. Soc. 43 (7) (2017), 2205-2226 [8] Estaji, A. A.; Robat Sarpoushi, M., On $$CP$$-frames, submitted [9] Estaji, A. A.; Robat Sarpoushi, M.; Elyasi, M., Further thoughts on the ring $$\mathcal{R}_c(L)$$ in frames, Algebra Universalis 80 (4) (2019), 14, https: //doi.org/10.1007/s00012-019-0619-z 4 [10] Ghadermazi, M.; Karamzadeh, O. A.S.; Namdari, M., On the functionally countable subalgebra of $$C(X)$$, Rend. Semin. Mat. Univ. Padova 129 (2013), 47-69, https://doi.org/10.4171/RSMUP/129-4 [11] Ghadermazi, M.; Karamzadeh, O. A.S.; Namdari, M., $$C(X)$$ versus its functionally countable subalgebra, Bull. Iranian Math. Soc. 45 (2019), 173-187, https://doi.org/10.1007/s41980-018-0124-8 [12] Gillman, L.; Jerison, M., Rings of Continuous Functions, Springer-Verlag, 1976 [13] Johnstone, P. T., Stone Spaces, Cambridge Univ. Press, Cambridge, 1982 [14] Karamzadeh, O. A.S.; Keshtkar, Z., On $$c$$-realcompact spaces, Quaest. Math. 42 (8) (2018), 1135-1167, https://doi.org/10.2989/16073606.2018.1441919 [15] Karamzadeh, O. A.S.; Namdari, M.; Soltanpour, S., On the locally functionally countable subalgebra of $$C(X)$$, Appl. Gen. Topol. 16 (2015), 183-207, https://doi.org/10.4995/agt.2015.3445 [16] Karimi Feizabadi, A.; Estaji, A. A.; Robat Sarpoushi, M., Pointfree version of image of real-valued continuous functions, Categ. Gen. Algebr. Struct. Appl. 9 (1) (2018), 59-75 [17] Mehri, R.; Mohamadian, R., On the locally countable subalgebra of $$C(X)$$ whose local domain is cocountable, Hacet. J. Math. Stat. 46 (6) (2017), 1053-1068, http://doi.org/10.15672/HJMS.2017.435 [18] Namdari, M.; Veisi, A., Rings of quotients of the subalgebra of $$C(X )$$ consisting of functions with countable image, Int. Math. Forum 7 (12) (2012), 561-571 [19] Picado, J.; Pultr, A., Frames and Locales: Topology without Points, Frontiers in Mathematics, Birkhäuser/Springer, Basel AG, Basel, 2012 [20] Robat Sarpoushi, M., Pointfree topology version of continuous functions with countable image, Hakim Sabzevari University, Sabzevar, Iran (2017), Phd. Thesis
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