Remarks on a result of Sibony on the Carathéodory topology

In this paper, we prove that if a Carathéodory hyperbolic analytic space $X$ is $C_X$-complete, then its natural topology is induced by the Carathéodory distance on $X$. This is an improvement of Sibony’s result, which concludes the same under the hypothesis that $X$ is $C_X$-finitely compact. This improvement is not merely formal; we also show the existence of uncountably many biholomorphically inequivalent analytic spaces that are not $C_X$-finitely compact but are $C_X$-complete.