Metrizability of the space of \(\mathbb{R}\)-places of a real function field
The topological space \(M_F\) of \(\mathbb R\)-places of the field \(F\) is a quotient space of the space \(X_F\) of all orderings of \(F.\) In general its structure can be very complicated. For \(F={\mathbb R}(X)\) the space \(M_F\) is the real projective line. However for \(n>1\) the structure of \(M_{{\mathbb R}(X_1,\dots,X_n)}\) is not known so far. The main theorem of the paper says that for any uncountable real closed field \(R\), the space of \(\mathbb R\)-real places of the rational function field \(R(X_1,X_2)\) is not metrizable. The authors also consider the more general question of when the space of \(\mathbb R\)-places of a finitely generated formally real field extension \(F\) of a real closed field \(R\) is metrizable. They prove that if \(R\) is uncountable and the transcendence degree of \(F\) over \(R\) is greater than 1, then \(M_F\) is not metrizable, whereas in the case of transcendence 1 and \(R\) being Archimedean the space \(M_F\) is metrizable if \(R\) is countable.
scientific article; zbMATH DE number 5782759
scientific article