A Note on Partial b-Metric Spaces

Abstract

Let (X,b) be a partial b-metric space with coefficient ${s \geq 1. \,\,{\rm For \,\, each}\,\, x \in X \,\,{\rm and \,\,each}\,\, \varepsilon > 0, \,\,{\rm put}\,\, B(x, \varepsilon) = \{y \in X \colon b(x,y) < b(x,x) + \varepsilon\} \,\,{\rm and \,\,put}\,\, \mathcal{B}=\{B(x,\varepsilon)\colon x\in X \, {\rm and} \,\, \varepsilon > 0\}}$. In this brief note, we prove that ${\mathcal{B}}$ is not a base for any topology on X, which shows that a claim on partial b-metric spaces is not true. However, ${\mathcal{B}}$ can be a subbase for some topology τ on X. For a sequence in X, we also give some relations between convergence with respect to τ and convergence with respect to b.