The Katětov dimension of proximity spaces. -- An internal approach (Q1364908)

Katětov's theorem [\textit{M. Katětov}, Časopis Mat. Fys. 75, 1-16 (1950; Zbl 0037.35402)] and [\textit{L. Gillman} and \textit{M. Jerison}, Rings of continuous functions (Zbl 0093.30001; Zbl 0327.46040)] characterizes the covering dimension of a completely regular space \(X\) in terms of the behaviour of certain subrings of the ring \(C^*(X)\); to be precise: \(\dim X\leq n\) if{}f every finitely generated analytic subring is contained in an analytic subring generated by no more than \(n\) elements, where a subring \(A\) is analytic if it contains every \(x\) that satisfies an equation of the form \(x^n+a_{n-1}x^{n-1}+\cdots a_1x+a_0=0\) with the coefficients \(a_i\) from \(A\). The authors adapt this result to the setting of proximity spaces and rings of proximally continuous functions. The proof is an adaptation of the exposition in Gillman and Jerison's book where all references to connected subsets of \(\beta X\) are replaced by so-called proximally fine filters -- these are the traces of neighbourhood filters of connected subsets of the Smirnov compactification.