Abstract:
Let $X$ and $Y$ be metrizable spaces. A map $X \overset{f}{\longrightarrow} Y$ is called topologically uniformly continuous, if for every admissible metric $\rho$ on $X$ there is an admissible metric $\sigma$ on $Y$ such that for the metric spaces $(X,\rho)$ and $(Y, \sigma)$ the map $(X,\rho) \overset{f}{\longrightarrow} (Y, \sigma)$ is uniformly continuous. In this article such maps are investigated. As the main result, it is shown that, in a certain sense, topologically uniformly continuous maps are close to perfect maps.
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