Some topological results on generalized parametric metric spaces
Abhishikta Das, T. Bag
TL;DR
This work extends metric space concepts to generalized parametric metric spaces $(X,P,o)$ by defining convergence, Cauchyness, and boundedness through $P$ and a binary operation $o$, and by constructing a topology $\tau_P$ from open balls $B(a,\alpha,t)$. It develops $\alpha$-metrics $d_\alpha$ and proves fundamental topological properties, including separation axioms, countability, and compactness notions, while introducing a diameter $\delta(A)$ and a Cantor-type completeness theorem. A key result is that nested closed sets with $\delta(A_i)\to0$ intersect in a single point, providing a generalized Cantor intersection framework, and that completeness follows from every Cauchy sequence having a convergent subsequence. Furthermore, when $o=\max$, the $\tau_P$ topology coincides with the $\tau_{d_\alpha}$ topology, recovering the classical metric-space topology within this generalized setting and linking the theory to standard topological results.
Abstract
In this paper, ideas of open ball, closed ball, compact set are introduced and some related basic properties are studied. Some topological properties and some other well known results of metric spaces including Cantor intersection theorem are established in generalized parametric metric space setting.