On e-spaces and rings of real valued e-continuous functions

Authors

DOI:

https://doi.org/10.4995/agt.2023.17743

Keywords:

e-space, e-continuous function, real maximal ideal, quasicomponent, zero-dimensional space

Abstract

Whenever the closure of an open set is also open, it is called e-open and if a space have a base consisting of e-open sets, it is called e-space. In this paper we first introduce and study e-spaces and e-continuous functions (we call a function f from a space X to a space Y an e-continuous at x ∈ X if for each open set V containing f(x) there is an e-open set containing x with f ( U ) ⊆ V ). We observe that the quasicomponent of each point in a space X is determined by e-continuous functions on X and it is characterized as the largest set containing the point on which every e-continuous function on X is constant. Next, we study the rings Ce( X ) of all real valued e-continuous functions on a space X. It turns out that Ce( X ) coincides with the ring of real valued clopen continuous functions on X which is a C(Y) for a zero-dimensional space Y whose elements are the quasicomponents of X. Using this fact we characterize the real maximal ideals of Ce( X ) and also give a natural representation of its maximal ideals. Finally we have shown that Ce( X ) determines the topology of X if and only if it is a zero-dimensional space.

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Author Biographies

S. Afrooz, Khorramshahr University of Marine Science and Technology

Faculty of Marine Engineering

F. Azarpanah, Shahid Chamran University of Ahvaz

Department of Mathematics

N. Hasan Hajee, Shahid Chamran University of Ahvaz

Department of Mathematics

References

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Z. Arjmandnezhad, F. Azarpanah, A. A. Hesari and A. R. Salehi, Characterizations of maximal $z^circ$-ideals of C(X) and real maximal ideals of q(X), Quaest. Math. 45 (2022), 1575-1587. https://doi.org/10.2989/16073606.2021.1966544

R. Engelking, General Topology, Heldermann Verlag, Berlin-west 31, 1989.

L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, 1976.

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S. Willard, General Topology, Addison-Wesley Publishing Company, Inc. 1970.

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Published

2023-10-02

How to Cite

[1]
S. Afrooz, F. Azarpanah, and N. Hasan Hajee, “On e-spaces and rings of real valued e-continuous functions”, Appl. Gen. Topol., vol. 24, no. 2, pp. 433–448, Oct. 2023.

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Section

Regular Articles