The space of \(\mathbb{R}\)-places on a rational function field (Q2204848)

Let \(F\) be a field and denote by \(\mathcal{M}(F)\) the set of all \(\mathbb{R}\)-places, i.e., places from \(F\) into the real numbers \(\mathbb{R}\). This set \(\mathcal{M}(F)\) becomes a topological space carrying the so-called Harrison topology with subbasis given by the Harrison sets \(H(a)=\{\tau\in \mathcal{M}(F)\,|\,0<\tau(a)<\infty\}\) (\(a\in F\)). The main theorem in this interesting paper states that if \(F\) is a field such that for each \(\mathbb{R}\)-place of \(F\), the value group of the corresponding valuation is trivial or countable, then the path-connected components of \(\mathcal{M}(F)\) are in natural bijection with the path-connected components of \(\mathcal{M}(F(x_1,\ldots,x_n))\), where \(F(x_1,\ldots,x_n)\) denotes the rational function field in \(n\) variables over \(F\). More precisely, for every \(n>0\), restriction of mappings induces a restriction map \(\mathit{res}:\mathcal{M}(F(x_1,\ldots,x_n)) \to \mathcal{M}(F)\). Taking preimages under \(\mathit{res}\) of path-connected components then yields the desired bijection. There are some nice consequences of the main theorem. For example, for every \(n\geq 0\), one obtains that \(\mathcal{M}(\mathbb{R}(x_1,\ldots,x_n))\) is path-connected. This had been proved earlier by the authors in [Ann. Math. Sil. 32, 99--131 (2018; Zbl 1407.12001)]. Also, if \(K\) is an algebraic extension of \(\mathbb{Q}\), then the number of path-connected components of \(\mathcal{M}(K(x_1,\ldots,x_n))\) is given by the number of homomorphisms of \(K\) into \(\mathbb{R}\). For example, \(\mathcal{M}(\mathbb{Q}(\sqrt{2})(x_1,\ldots,x_n))\) has two path connected-components for any \(n\geq 0\).