Remarks on the rings of functions which have a finite numb er of di scontinuities

• Ahmadi Zand, Mohammad Reza ^[1] ; Khosravi, Zahra ^[1]
  1. [1] Yazd University
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 22, Nº. 1, 2021, págs. 139-147
• Idioma: inglés
• DOI: 10.4995/agt.2021.14332
• Enlaces
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• Resumen

  • 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.

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