On rings of real valued clopen continuous functions
Keywords:clopen continuous (cl-supercontinuous), zero-dimensional, Ps-space, almost Ps-space, Baer ring, p.p. ring, quasi-component, socle, mildly compact, s-basically and s-extremally disconnected space
Among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper. We investigate and study the ring Cs(X) of all real valued clopen continuous functions on a topological space X. It is shown that every ƒ âˆˆ Cs(X) is constant on each quasi-component in X and using this fact we show that Cs(X) â‰… C(Y), where Y is a zero-dimensional s-quotient space of X. Whenever X is locally connected, we observe that Cs(X) â‰… C(Y), where Y is a discrete space. Maximal ideals of Cs(X) are characterized in terms of quasi-components in X and it turns out that X is mildly compact(every clopen cover has a finite subcover) if and only if every maximal ideal of Cs(X)is fixed. It is shown that the socle of Cs(X) is an essential ideal if and only if the union of all open quasi-components in X is s-dense. Finally the counterparts of some familiar spaces, such as Ps-spaces, almost Ps-spaces, s-basically and s-extremally disconnected spaces are defined and some algebraic characterizations of them are given via the ring Cs(X).
S. Afrooz, F. Azarpanah and O. A. S. Karamzadeh, Goldie dimension of rings of fractions of C(X), Quaest. Math. 38, no. 1 (2015), 139-154. https://doi.org/10.2989/16073606.2014.923189
F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125, no. 7 (1997), 2149-2154. https://doi.org/10.1090/S0002-9939-97-04086-0
F. Azarpanah, On almost P-spaces, Far East J. Math. Sci. Special volume (2000), 121-132.
F. Azarpanah and O. A. S. Karamzadeh, Algebraic characterizations of some disconnected spaces, Italian J. Pure Appl. Math. 12 (2002), 155-168.
R. Engelking, General Topology, Sigma Ser. Pure Math., Vol. 6, Heldermann Verlag, Berlin, 1989.
L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, 1976.
O. A. S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184. https://doi.org/10.2307/2044578
R. Levy, Almost P-spaces, Can. J. Math. XXIX, no. 2 (1977), 284-288. https://doi.org/10.4153/CJM-1977-030-7
I. L. Reilly and M. K. Vamanamurthy, On supercontinuous mappings, Indian J. Pure Appl. Math. 14, no. 6 (1983), 767-772.
D. Singh, cl-supercontinuous functions, Applied Gen. Topol. 8, no. 2 (2007), 293-300. https://doi.org/10.4995/agt.2007.1899
R. Staum, The algebra of bounded continuous functions into non-archimedean field, Pacific J. Math. 50, no. 1 (1974), 169-185. https://doi.org/10.2140/pjm.1974.50.169
S. Willard, General Topology, Addison-Wesley Publishing Company, Inc., 1970
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