Remarks on the rings of functions which have a finite numb er of di scontinuities
Submitted: 2020-09-14
|Accepted: 2020-12-25
|Published: 2021-04-01
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Keywords:
C(X)F, Z-ultrafilter, completely separated, C(X)F -embedded, Z-filter, over-rings of C(X), Artinian ring
Supporting agencies:
Department of pure Mathematics Yazd university
Abstract:
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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