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+O_{x}) 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)=C^{F}(X) , where C^{F}(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)=C_{C}(X), where C_{C}(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.

**Keywords**

**Main Subjects**

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April 2021

Pages 40-48

**Receive Date:**25 December 2019**Revise Date:**26 July 2020**Accept Date:**11 September 2020**First Publish Date:**19 January 2021