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On convergence of closed sets in a metric space and distance functions. (English) Zbl 0558.54007
Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence $$\{C_ n\}$$ in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

##### MSC:
 54B20 Hyperspaces in general topology 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 40A30 Convergence and divergence of series and sequences of functions 54C35 Function spaces in general topology
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