Uniformly continuous mappings between quasimetric spaces are studied and a topological homeomorphism between two compact Hausdorff partially metric spaces is constructed such that the mapping between the corresponding quasimetric spaces is not uniformly continuous. This example, in particular, shows that Theorem 4.4 from [6] is false. In addition, an analogue of the Heine-Cantor theorem on the uniform continuity of an arbitrary continuous mapping $f:X\to Y$ defined on a premetric space $X$ that satisfies some strengthened condition of countable compactness and takes values ​​in the uniform space $Y$ is proved. An example of a continuous mapping $f:X\to Y$ defined on a compact Hausdorff premetric space $X$ and with values ​​in a uniform space $Y$ that is not uniformly continuous is also given.

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