A Note on Middle P-spaces and Related Rings

Abstract

A zero-set of a Tychonoff space is called a middle zero-set if it is the intersection of two proper zero-sets the union of which is the whole space. If every nonempty middle zero-set has nonempty interior, then the space is called a middle P-space. These spaces were introduced by Azarpanah and Motamedi in their study of the zero-divisor graph of rings of continuous functions. In this note, we explore some topological properties of these spaces. We also characterize them in terms of their rings of continuous functions. We apply the methods of pointfree topology for purposes of lucidity and breadth of scope.