Article

Some new results on functions in C(X) having their support on ideals of closed sets

Published in: Quaestiones Mathematicae
Volume 42 , issue 8 , 2019 , pages: 1079–1090

DOI: 10.2989/16073606.2018.1504830

Author(s): Sudip Kumar Acharyya Department of Pure Mathematics, India , Sagarmoy Bag Department of Pure Mathematics, India , Goutam Bhunia Department of Pure Mathematics, India , Pritam Rooj NIIT University, India

Keywords: Mathematics Subject Classification (2010): Primary 54C40 , Secondary 46E25 , Mathematics Subject Classification (2010): Primary 54C40 , Secondary 46E25

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Abstract

For any ideal of closed sets in X, let be the family of those functions in C(X) whose support lie on . Further let contain precisely those functions f in C(X) for which for each ϵ > 0, {x ∈ X: |f (x)| ≥ ϵ} is a member of . Let stand for the set of all those points p in βX at which the stone extension f^∗ for each f in is real valued. We show that each realcompact space lying between X and βX is of the form if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally- or almost locally-, becomes a space of the same form. We further show that is a free ideal (essential ideal) of C(X) if and only if is a free ideal (essential ideal) of when and only when X is locally- (almost locally-). We address the problem, when does or become identical to the socle of the ring C(X). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice ≡ the ideal of compact sets in X. Finally we observe that the ideals of the form of C(X) are no other than the z^◦-ideals of C(X).

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