Locally constant functions and oc-paracompact spaces

Document Type : Original Paper

Author

Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Abstract

In this article we investigate and study the ring LC(X) of all real-valued locally constant functions on a topological space X . We show that X is a connected space if and only if LC(X)=R. If X is a compeletly regular and Hausdorff space, we show that LC(X) is always Von Neumann regular ring and also we prove that LC(X)=∩{xin N}(R+Ox) which N is the set of all non-isolated points of X . Also we show that X is a P-space if and only if LC(X)=C(X), where C(X) denotes the ring of all real-valued continuous functions . It is also shown that X is a weakly pseudocompact space if and only if LC(X)=CF(X) , where CF(X) denotes the ring of all real-valued continuous functions with finite image. In case X is Lindel of, we prove that it is a CP-space if and only if LC(X)=CC(X), where CC(X) denotes the ring of all real-valued continuous functions with countable image. We introduce the concept of "oc-paracompact" and we observe that an oc-paracompact space is compact if and only if it is weakly pseudocompact. Finally, we show that if X is a zero dimensional and second countable space , then X is compact if and only if it is a weakly pseudocompact space.

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References

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Journal of Advanced Mathematical Modeling
Volume 11, Issue 1
April 2021
Pages 40-48

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History

  • Receive Date: 25 December 2019
  • Revise Date: 26 July 2020
  • Accept Date: 11 September 2020

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