Remarks on a result of Sibony on the Carathéodory topology

TL;DR

The paper strengthens Sibony’s criterion by showing that Carathéodory-completeness alone suffices for the Carathéodory distance to induce the natural topology on a Carathéodory hyperbolic space, removing the need for finite-compactness. It develops a robust topological-analytic framework and builds explicit, uncountably many examples X(R) that are Carathéodory-complete but not C_X-finitely compact, yet share the same Carathéodory topology as their natural topology. The combination of a general equivalence lemma and concrete constructions demonstrates both the theoretical improvement and the abundance of spaces satisfying top(X) = top(C_X). The results have implications for understanding when metric and topological structures align in complex analysis.

Abstract

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.

Theorems & Definitions (19)

• Definition 1.1
• Remark 1.2
• Remark 1.3
• Definition 1.4
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof
• Definition 3.1