Some separation axioms in topoframes

Document Type : Research Paper


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

2 Department of Mathematics, Islamic Azad University (Gorgan Branch), Gorgan, Iran



This paper is about the extension of some classical separation axioms Hausdorffness, regularity and complete regularity to topoframes. We show that they agree with those in frames except perhaps for complete regularity. The interesting results are about complete regularity, in particular when and how these differ from the frame results. These together with the results about B-filters are the focus of the paper.


[1] A. R. Aliabad and A. Sheykhmiri, LG-Topology, Bull. Iranian Math. Soc., 41 No. 1 (2015) 239-258.
[2] A. R. Aliabad and H. Zarepour, Continuous Functions on LG-spaces, J. Algebr. Syst., 8 No. 2 (2021) 181-200.
[3] R. N. Ball and A. W. Hager, On the localic Yosida representation of an archimedean lattice ordered group with weak unit, J. Pure Appl. Algebra, 70 (1991) 17-43.
[4] R. N. Ball and J. Walters-Wayland, C- and C*-quotients in pointfree topology, Dissertationes Math. (Rozprawy Mat.), 412 (2002) 1-61.
[5] B. Banaschewski, The real numbers in pointfree topology, Textos Mat. Sér. B, 12 (1997) 1-96.
[6] E. Čech, On bicompact spaces, Ann. of Math., 38 (1937) 823-844.
[7] T. Dube, A note on the socle of certain types of f-rings, Bull. Iranian Math. Soc., 38 No. 2 (2012) 517-528.
[8] T. Dube, Extending and contracting maximal ideals in the function rings of pointfree topology, Bull. Math. Soc. Sci. Math. Roumanie, (2012) 365-374.
[9] T. Dube, Some algebraic characterizations of F-frames, Algebra Univers., 62 (2009) 273-288.
[10] T. Dube, Some ring-theoretic properties of almost P-frames, Algebra Univers., 60 (2009) 145-162.
[11] A. A. Estaji and M. Abedi, Some ring-theoretic properties of the ring of R(Lτ), (Accepted).
[12] A. A. Estaji, J. Farokhi ostad and T. Haghdadi,Weight of Boolean I-contact algebra, Bull. Iranian Math. Soc., (2021) 1-15.
[13] A. A. Estaji, T. Haghdadi, and J. Farokhi-Ostad, Topobooleans and boolean contact algebras with interpolation property, Filomat, 35 No. 9 (2021) 2895-2909.
[14] A. A. Estaji, A. Karimi Feizabadi and M. Zarghani, The ring of real-continuous functions on a topoframe,Categ. Gen. Algebr. Struct. Appl., 4 No. 1 (2016) 75-94.
[15] A. A. Estaji, A. Karimi Feizabadi and M. Zarghani, Zero elements and z-ideals in modified pointfree topology, Bull. Iranian Math. Soc., 43 No. 7 (2017) 2205-2226.
[16] A. A. Estaji and M. Taha, z-ideals in the real continuous function ring R(Lτ ), Bull. Iranian Math. Soc., 46 (2020) 37-51.
[17] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, New York, NY, 1976.
[18] S. Givant and P. Halmos, Introduction to Boolean Algebras, Springer+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA, 2007.
[19] W. Guo-Jun, Topological molecular lattices (I), Kexue Tongbao,29 (1984) 19-23.
[20] W. Guo-Jun, Theory of topological molecular lattices, Fuzzy Sets Syst., 47 (1992) 351-376.
[21] P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982.
[22] J. Paseka and B. Šmarda, T2-frames and almost compact frames, Czechoslovak Math. J., 42 No. 3 (1992) 385-402.
[23] J. Picado and A. Pultr, Frames and Locales: Topology without points, Springer Science & Business Media, 2012.
[24] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937) 375-481.
[25] A. Tychonoff, Über die topologische Erweiterung von Raümen, Math. Ann., 102 (1930) 544-561.
[26] M. Zarghani, The ring of real-continuous functions on a topoframe, PhD thesis, Hakim Sabzevari University, 2017.
[27] M. Zarghani, A. A. Estaji, and A. Karimi Feizabadi, A new pointfree form of topology, 47th Annual Iranian Mathematics Conference (AIMC47), Kharazmi University, Karaj, Iran, 2016.
[28] S. Willard, General Topology, Addison-Wesley Publishing Company Inc., 1970.