On the largest regular subring of C(X)

Abstract

We call a real-valued function f on a topological space X a locally constant if each point x ∈ X has a neighborhood on which f is constant. The set of all locally constant functions f: X → ℝ is denoted by E(X). We observe that E(X) is the largest von Neumann regular subring of C(X) containing ℝ. We show that for every space X, there is a zero-dimensional space Y such that E(X) ≅ E(Y) and E(X) determines the topology of X if and only if X is a zero-dimensional space. By an example, it turns out that E(X) is not always of the form C(Y) for some space Y, but whenever X is a locally connected space, we observe that E(X) is a C(Y), where Y is a discrete space. Finally some von Neumann local subrings of C(X) containing ℝ are introduced and topological spaces X are characterized for which these subrings coincide with the smallest one. Note that a ring R is said to be a von Neumann local ring if for each a ∈ R either a or 1 − a has von Neumann inverse.