Remarks on a result of Sibony on the Carath\'eodory topology

TL;DR

This paper improves Sibony's result by showing that Carathéodory hyperbolic spaces that are C_X-complete have their topology induced by the Carathéodory distance, and demonstrates the existence of many non-compact yet complete examples.

Contribution

It extends Sibony's theorem from C_X-finite compactness to C_X-completeness and constructs numerous examples of such spaces that are not finitely compact.

Findings

C_X-complete spaces have topology induced by Carathéodory distance

Existence of uncountably many non-finitely compact but C_X-complete spaces

Improvement over previous results requiring finite compactness

Abstract

In this paper, we prove that if a Carath\'eodory hyperbolic analytic space $X$ is $C_X$-complete, then its natural topology is induced by the Carath\'eodory 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.