A minimal regular ring extension of \(C(X)\) (Q2773373)

A commutative ring \(B\) is said to be (von Neumann) regular in case each \(b\in B\) has a unique quasi-inverse \(b^*\) such that \(b^2b^*= b\) and \((b^*)^2b= b^*\). Let \(X\) be a Tikhonov space with topology \(\tau\), \(F(X)\) be the ring of all real valued functions on \(X\), and \(C(X,\tau)\) the subring of those elements continuous in \(\tau\). Then \(F(X)\) is a regular ring, and the smallest regular subring containing \(C(X,\tau)\) is denoted by \(G(X)\). If \(\tau_{\delta}\) denotes the topology generated by the \(G_{\delta}\)-sets of \(\tau\) as an open base, then \(C(X,\tau_{\delta})\) is a regular ring and \(C(X,\tau)\subset G(X)\subset C(X,\tau_{\delta})\).NEWLINENEWLINENEWLINEIn the present paper, the authors call \(X\) an RG-space in case \(G(X)= C(X,\tau_{\delta})\) and investigate when this condition is satisfied. Their results includeNEWLINENEWLINENEWLINE(a) a subspace \(Y\) of an RG-space is an RG-space in case \(Y\) is scattered and Lindelöf;NEWLINENEWLINENEWLINE(b) countable subsets of RG-spaces are scattered;NEWLINENEWLINENEWLINE(c) compact subspaces of RG-spaces are scattered RG-spaces;NEWLINENEWLINENEWLINE(d) if \(P(X)\) denotes the set of P-points of \(X\) and \(X- P(X)\) is finite, then \(X\) is an RG-space.NEWLINENEWLINENEWLINELet \(D_1(X)\) be the derived set of \(X\) and define \(D_\alpha(X)\) iteratively by transfinite induction for each ordinal \(\alpha\). If \(X\) is scattered, then \(D_\alpha(X)= \emptyset\) for some \(\alpha\), and the least of such \(\alpha\) is called by the authors the Cantor-Bendixson order \(CB(X)\). Their main result is that if \(X\) is compact or metric, then \(X\) is an RG-space if, and only if, \(X\) is scattered with \(CB(X)\) finite. They provide a number of counterexamples and raise several open questions.