Some Remarks on Partial Metric Spaces
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In this paper, we investigate some topological properties of partial metric spaces (in short PMS). We give some relationship between metric-like PMS, sequentially isosceles PMS and sequentially equilateral PMS. We also prove a type of Urysohn’s lemma for metric-like PMS. By applying the construction of Hartman–Mycielski, we show that every bounded PMS can be isometrically embedded into a pathwise connected and locally pathwise connected PMS. In the end, we show that a partial metric space is compact iff it is totally bounded and complete.
KeywordsPartial metric spaces Urysohn’s lemma Isometric embedding Compact space Complete space
Mathematics Subject ClassificationPrimary 54E35 54E50 Secondary 54D05 54D30
The authors are grateful to the referees whose comments and observations have improved the presentation of the article. Wei He acknowledges the support of NSFC Grant No. 11571175.
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