On the locally countable subalgebra of C(X) whose local domain is cocountable

Abstract

In this paper, we present a new subring of $C(X)$ that contains the subring $C_c(X)$, the set of all continuous functions with countable image. Let $L_{cc}(X)=\{ f\in C(X)\,:\, |X\backslash C_f|\leq \aleph_0 \}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq \aleph_0$. We observe that $L_{cc}(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$. It is shown that any hereditary lindel\"{o}f scattered space is functionally countable.Spaces $X$ such that $L_{cc}(X)$ is regular (von Neumann) are characterized and it is shown that $\aleph_0$-selfinjectivity and regularity of $L_{cc}(X)$ coincide.

Keywords

• functionally countable space,uncountable-open space,socle,locally c-scattered space,$\aleph_0$-selfinjective

References

1. Atiyah, M. F. and Macdonald, I. G. Introduction to commutative algebra, Addison-Wesley, Reading, Mass., 1969.
2. Azarpanah, F. Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125, 2149- 2154, 1997.
3. Azarpanah, F. and Karamzadeh, O. A. S. Algebraic characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12, 155-168, 2002.
4. Azarpanah, F., Karamzadeh, O. A. S. and Rahmati, S. C(X) VS. C(X) modulo its socle, Colloq. Math. 3, 315-336, 2008.
5. Bhattacharjee, P., Knox, M. L. and Mcgovern, W. WM. The classical ring of quotients of Cc(X), Appl. Ge Topol. 15, no. 2, 147-154, 2014.
6. Azarpanah, F., Manshoor, F. and Mohamadian, R. Connectedness and compactness in C(X) with the m-topology and generalized m-topology, Topology and its applications 159, 3486-3493, 2012.
7. Dovgoshey, O., Martio, O., Ryazanov, V. and Vuopinen, M. The Cantor function, Expo. Math. 24, 1-37, 2006.
8. Dube, T. Contracting the socle in rings of continuous functions, Rend. Semin. Mat. Univ. Padova 123, 37-53, 2010.

9. Engelking, R. General topology, Heldermann Verlag Berlin, 1989.
10. Estaji, A. A. and Karamzadeh, O. A. S. On C(X) modulo its socle, Comm. Algebra 31, 1561-1571, 2003.
11. Ganster, M. A note on strongly lindelöf spaces, Soochow J. Math. 15 (1), 99-104, 1989.
12. Ghadermazi, M., Karamzadeh, O. A. S. and Namdari, M. On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova, 129, 47-69, 2013.
13. Ghadermazi, M., Karamzadeh, O. A. S. and Namdari, M. C(X) versus its functionally countable subalgebra, submitted in 2013.
14. Ghadermazi, M. and Namdari, M. On -scattered spaces, Far East J. Math. Sci.(FJMS), 32 (2), 267-274, 2009.
15. Ghasemzadeh, S. G., Karamzadeh, O. A. S. and Namdari, M. The super socle of the ring of continuous functions, Mathematica Slovaka, to appear.
16. Gillman, L. and Jerison, M., Rings of continuous functions, Springer-Verlag, 1976.
17. Goodearl, K. R., Von Neumann Regular Rings, Pitman, 1979.
18. Henriksen, M., Raphael, R. and Woods, R. G. SP-scattered spaces; a new generalization of scattered spaces, Comment. Math. univ. carolin 48, no. 3, 487–505, 2007.
19. Kaplansky, I. Commutative rings, Boston: Allyn and Bacon, Inc, 1970.
20. Karamzadeh, O. A. S. On a question on Matlis, Comm. Algebra, 25, 2717-2726, 1997.
21. Karamzadeh, O. A. S. and Koochakpour, A. A. On $\aleph_0$-selfinjectivity of strongly regular rings, Comm. Algebra 27, 1501-1513, 1999.
22. Karamzadeh, O. A. S. and Motamedi, M. A note on rings in which every maximal ideal is generated by a central idempotent, Proc. Japan Acad., 58 Ser. A, 124-125, 1982.
23. Karamzadeh, O. A. S., Namdari, M. and Soltanpour, S. On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol., 16, no (2) 183-207, 2015.
24. Karamzadeh, O. A. S. and Rostami, M. On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93, 179-184, 1985.
25. Lambek, J. Lectures on rings and modules, Blaisdell, 1966.
26. Levy, R. and Rice, M.D. Normal P-spaces and the G-topology, Colloq. Math. 47, 227-240, 1981.
27. Namdari, M. and Veisi, A. The subalgebra of Cc(X) consisting of elements with countable image versus C(X) with respect to their rings of quotients, Far East J. Math. Sci. (FJMS), 59, 201-212, 2011.
28. Pelczynski, A. and Semadeni, Z. Spaces of continuous functions (III), Studia Math. 18, 211-222, 1959.
29. Rudd, D. On two sum theorems for ideals of C(X), Michigan Math. J. 17, 139-141, 1970.
30. Rudin, W. Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8, 39-42, 1957.
31. Rudin, M. E. and Rudin, W. Continuous functions that are locally constant on dense sets, J. Funct. Anal. 133, 120-137, 1995.
32. Willard, S. General topology, Addison-Wesley, Reading, MA, 1970.