On the extension of partial metrics and quasi-metrics

In a recent article, V. Mykhaylyuk and V. Myronyk investigated the problem of extending partial metrics and quasi-metrics. Concretely, they stated, and proved, as a main result that if A is a closed subset of a partially metrizable space (X,τ) then every compatible partial metric p on A whose induced quasi-metric qp is bounded admits an extension to a compatible partial metric on X. They also gave an example showing that the boundedness of the quasi-metric qp cannot be omitted. Since the partially metrizable space of their example is not T1, the authors raised the natural question of whether the boundedness of the quasi-metric qp can be removed when the partially metrizable space is T1 . Moreover, they also posed a similar question in the framework of quasi-metrizable spaces. In this note we present two examples showing that Mykhaylyuk and Myronyk’s questions have a negative answer.