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The aim is to explore the properties of metric spaces that lack the symmetry axiom, with a focus on completeness and convergence-symmetry.

June 20, 2026Open Access

Convergence-Symmetric Metric Spaces

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The aim is to explore the properties of metric spaces that lack the symmetry axiom, with a focus on completeness and convergence-symmetry.
Discussed examples of asymmetric metrics such as Thurston's metric, earthquake metric, Funk metric, Apollonian metric, and left Hausdorff metric.
Proved results including a Banach fixed point theorem, an Arzelà–Ascoli theorem, and a Hopf–Rinow theorem in an asymmetric context.
Introduced the concept of convergence-symmetry and compared it to Busemann's condition.
Every convergence-symmetric metric space has a unique minimal completion that is also convergence-symmetric.
Several examples met Busemann's condition but were not convergence-symmetric, highlighting the distinct nature of convergence-symmetry.
Established new results relevant to the study of asymmetric metrics, indicating further research opportunities.

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Demonstrates properties of asymmetric metrics in metric spaces, indicating the need for new theories.

Saglam et al. (2026) studied this question.

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