Uniform Continuity in Fuzzy Quasi-Metric Spaces: Foundations, Extensions, and Structural Properties

Resumen

This paper explores uniform continuity in fuzzy quasi-metric spaces, which generalize classical metric spaces by allowing asymmetry and fuzziness. The authors distinguish between pointwise and uniform continuity in these spaces and extend classical continuity results to the fuzzy setting. Key contributions include an analysis of how uniform continuity behaves under composition, inversion, and symmetrization, and how it is preserved under uniform equivalence of fuzzy quasi-metrics. A central result is an Extension Theorem, showing that any uniformly continuous function defined on a dense subspace of a complete fuzzy quasi-metric space can be uniquely extended while preserving uniform continuity. The paper also shows that uniformly continuous functions preserve fuzzy Cauchy sequences and fuzzy total boundedness, connecting these concepts to convergence and compactness in fuzzy environments. While uniform continuity is retained under restriction, the asymmetry of the space complicates inverse mappings. Examples and counterexamples illustrate the nuanced behavior of such functions.