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Metric spaces with nice closed balls and distance functions for closed sets. (English) Zbl 0588.54014
A metric space $$<X,d>$$ is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation $$F=Lim F_ n$$ for sequences of closed sets is equivalent to the pointwise convergence of $$<d(\cdot,F_ n)>$$ to d($$\cdot,F)$$. We also reconcile these modes of convergence with three other closely related ones.

MSC:
 54B20 Hyperspaces in general topology 54E35 Metric spaces, metrizability
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