Metrizability of some analytic affine spaces (Q2751736)

Let \(\mathbb{K}\) be a complete ultrametric field, \(A\) a \(\mathbb{K}\)-algebra and \(M(A)\) denotes the set of all multiplicative \(\mathbb{K}\)-algebra semi-norms on \(A\). If \(A\) is normed, the set \(M(A,\|\cdot\|)\) called the multiplicative spectrum of \(A\), is the set of \(\varphi\in M(A)\) which are continuous with respect to the norm \(\|\cdot\|\) on \(A\). For any \(n\in\mathbb{N}\), \(\mathbb{A}^n\) denotes the analytic affine space \(M(\mathbb{K}[x_1,x_2,\dots, x_n])\) of dimension \(n\) endowed with the topology of simple convergence.NEWLINENEWLINENEWLINEIn this paper it is shown that if \(\mathbb{K}\) is topologically separable (i.e. \(\mathbb{K}\) contains a dense countable subfield), then the topology of simple convergence of \(\mathbb{A}^n\) is metrizable. As a direct consequence of the metrizability of \(\mathbb{A}^n\), the arcwise connectedness of \(\mathbb{A}^n\) and the sequential compactness of the multiplicative spectrum of Tate \(\mathbb{K}\)-algebras are derived.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].