Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,Τ). We investigate when G(X) coincides with the ring C(X,Τδ) of continuous real-valued functions on the space (X,Τδ), where Τδ is the smallest Tikhonov topology on X for which tau subset of or equal to tau(delta) and C(X,Τδ) is von Neumann regular. The compact and metric spaces for which G(X) = C(X,Τδ) are characterized. Necessary, and different sufficient, conditions for the equality to hold more generally are found.
© 2002 Institute of Mathematics, Polish Academy of Sciences
Henriksen, M., R. Raphael, and R. G. Woods. "A minimal regular ring extension of C(X)." Fundamenta Mathematicae 172.1 (2002): 1-17.