Complex Analytic Manifolds without Countable Base

1. It has been shown by Radó3 that every Riemann surface satisfies Hausdorff's second countability axiom. In the same paper Radó gives a construction due to Prüfer of a real 2-dimensional locally Euclidean space whose open sets do not have a countable base. The purpose of this paper is to confirm a conjecture of Bochner that there exist non second-countable complex manifolds of complex dimension »> 1. The authors have independently constructed examples of such manifolds by two different methods, the first using the local quadratic transformations of algebraic geometry, the second by verifying that a modification of Prüfer's example has the structure of a 2-dimensional real analytic manifold that can be extended into the complex domain by formally replacing real parameters by complex ones. We describe these manifolds, the relations among them, and some of their topological properties.