On Topologically Uniformly Continuous Maps

Let $$X$$ and $$Y$$ be metrizable spaces. A map $$X\;\mathop \to \limits^f \;Y$$ is called topologically uniformly continuous if for every admissible (i.e., consistent with topology) metric $$\rho $$ on $$X$$ there exists an admissible metric $$\sigma $$ on $$Y$$ such that the map $$(X,\rho )\;\mathop \to \limits^f \;(Y,\sigma )$$ is uniformly continuous for the metric spaces $$(X,\rho )$$ and $$(Y,\sigma )$$ . This paper analyzes such maps. The main result is that, in a certain sense, topologically uniformly continuous maps are close to perfect maps.