Maximal independent, analytic sets in abelian Polish groups (Q2905568)

Sierpiński proved that a Hamel basis of \(\mathbb R\) over \(\mathbb Q\) cannot be analytic (a fortiori, cannot be Borel). Moreover, Bartoszyński et al. extended this result in an infinite dimensional separable real Banach space.NEWLINENEWLINEStarting from Sierpiński's original idea, the authors prove the following facts:NEWLINENEWLINE(1) Let \(K\) be a Polish field endowed with a nondiscrete ring topology and \(L\) a dense analytic proper subfield of \(K\). Then no basis of \(K\) as a vector space over \(L\) can be analytic.NEWLINENEWLINE(2) Let \(G\) be a nondiscrete topological Polish Abelian group. Then no maximal independent subset of \(G\) can be analytic.NEWLINENEWLINE(3) Let \(L\) be a subfield of a topological field \(K\). Assume that \(K\) is a nondiscrete Polish space and \(L\) is a nondiscrete subspace of \(K\). If \(K| L\) is a trascendental extension, then no trascendence basis of \(K| L\) can be analytic.NEWLINENEWLINENEWLINENEWLINE Recall that a Polish space is a topological space that is homeomorphic to a separable complete metric space. An analytic space is a Hausdorff topological space that is a continuous image of a Polish space.