An example of idiosyncracy between topological coverings and analytic functions

We exhibit a locally biholomorphic mapping with simply connected image, which fails to be a topological covering. Introduction In [1], A.F.Beardon shows an example of a ’covering surface’ over the unit disc D ⊂ C, which is not a topological covering. We recall that a ’covering surface’ of a regionD ⊂ C, is a Riemann surface S admitting surjective conformal mapping onto D, whereas a continuous mapping p of a topological space Y onto another one X is a ’topological covering’ if each point x ∈ X admits an open neighbourhood U such that the restriction of p to each connected component Vi of p−1(U) is a homeomorphism of Vi onto U . We also recall that a continuous mapping p of a topological space Y onto another one X has the ’curve-lifting’ property if for every curve γ : I → X and every y ∈ p−1(γ(0)) there exists a curve γ̃ : I → Y such that p ◦ γ̃ = γ and γ̃(0) = y. It is a standard topological result (see e.g. [3], section 9.3 and [2], theorem 4.19) that ’a local homeomorphism of a topological space Y onto another one X is a topological covering if and only if it has the curve-lifting property’. Beardon’s example takes origin by analytically continuing a holomorphic germ, namely a branch of the inverse of the Blaschke product