Derived Analytic Geometry for $$\mathbb {Z}$$ -Valued Functions Part I: Topological Properties

We study the Banach algebras \(\textrm{C}(X, R)\) of continuous functions from a compact Hausdorff topological space X to a Banach ring R whose topology is discrete. We prove that the Berkovich spectrum of \(\textrm{C}(X, R)\) is homeomorphic to \(\zeta (X) \times \mathscr {M}(R)\), where \(\zeta (X)\) is the Banaschewski compactification of X and \(\mathscr {M}(R)\) is the Berkovich spectrum of R. We study how the topology of the spectrum of \(\textrm{C}(X, R)\) is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of \(\zeta (X)\) can be easily reconstructed from the homotopy Zariski topology associated with \(\textrm{C}(X, R)\). We also prove some results about the existence of Schauder bases on \(\textrm{C}(X, R)\) and a generalization of the Stone–Weierstrass Theorem, under suitable hypotheses on X and R.