Article
Full entry |
PDF
(0.6 MB)
Feedback
PDF
(0.6 MB)
Feedback
Keywords:
functionally countable subalgebra; locally functionally countable subalgebra; sublocale; frame
functionally countable subalgebra; locally functionally countable subalgebra; sublocale; frame
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.
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 DOI 10.1216/RMJ-2018-48-2-345 | MR 3809150
[2] Ball, R.N., Walters-Wayland, J.: $\text{C}$- and $\text{C}^*$- quotients in pointfree topology. Dissertationes Math. (Rozprawy Mat.) 412 (2002), 354–384. MR 1952051
[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. MR 1621835 | Zbl 0891.54009
[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 DOI 10.4995/agt.2014.3181 | MR 3267269
[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. MR 0238744
[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 DOI 10.2298/FIL1819741E | MR 3899307
[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. MR 3885660
[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. DOI 10.1007/s00012-019-0619-z | MR 4027118
[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 DOI 10.4171/RSMUP/129-4 | MR 3090630
[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 DOI 10.1007/s41980-018-0124-8 | MR 3913987
[12] Gillman, L., Jerison, M.: Rings of Continuous Functions. Springer-Verlag, 1976. MR 0407579 | Zbl 0327.46040
[13] Johnstone, P.T.: Stone Spaces. Cambridge Univ. Press, Cambridge, 1982. MR 0698074 | Zbl 0499.54001
[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 DOI 10.2989/16073606.2018.1441919 | MR 3885948
[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 DOI 10.4995/agt.2015.3445 | MR 3411461
[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. MR 3833111
[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 DOI 10.15672/HJMS.2017.435 | MR 3751773
[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. MR 2969547
[19] Picado, J., Pultr, A.: Frames and Locales: Topology without Points. Frontiers in Mathematics, Birkhäuser/Springer, Basel AG, Basel, 2012. MR 2868166
[20] Robat Sarpoushi, M.: Pointfree topology version of continuous functions with countable image. Hakim Sabzevari University, Sabzevar, Iran (2017), Phd. Thesis.
Search
Partner of
