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.
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References
Shukla S.: Partial b-metric spaces and fixed point theorems. Mediterr. J. Math. 75, 3210–3217 (2012)
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Ge, X., Lin, S. A Note on Partial b-Metric Spaces. Mediterr. J. Math. 13, 1273–1276 (2016). https://doi.org/10.1007/s00009-015-0548-9
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DOI: https://doi.org/10.1007/s00009-015-0548-9