Remarks on the rings of functions which have a finite numb er of di scontinuities
Let X be an arbitrary topological space. F(X) denotes the set of all real-valued functions on X and C(X)F denotes the set of all f ∈ F(X) such that f is discontinuous at most on a finite set. It is proved that if r is a positive real number, then for any f ∈ C(X)F which is not a unit of C(X)F there exists g ∈ C(X)F such that g ≠ 1 and f = gr f. We show that every member of C(X)F is continuous on a dense open subset of X if and only if every non-isolated point of X is nowhere dense. It is shown that C(X)F is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein.
M. R. Ahmadi Zand, An algebraic characterization of Blumberg spaces, Quaest. Math. 33, no. 2 (2010), 223-230. https://doi.org/10.2989/16073606.2010.491188
A. J. Berrick and M. E. Keating, An Introduction to Rings and Modules, Cambridge University Press, 2000. https://doi.org/10.1017/9780511608674
W. Dunham, T1/2 -spaces, Kyungpook Math. J. 17, no. 2 (1977), 161-169.
R. Engelking, General Topology, Sigma Ser. Pure Math. 6, Heldermann-Verlag, Berlin, 1989.
Z. Gharabaghi, M. Ghirati. and A. Taherifar, On the rings of functions which are discontinuous on a finite set, Houston J. Math. 44, no. 2 (2018), 721-739.
L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York-Heidelberg, 1976.
N. Levine, Generalized closed sets in topology. Rend. Circ. Mat. Palermo. 19, no. 2 (1970), 89-96. https://doi.org/10.1007/BF02843888
Metrics powered by PLOS ALM
Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Universitat Politècnica de València
e-ISSN: 1989-4147 https://doi.org/10.4995/agt