Abstract:
We consider the space
${\rm clos}(X)$ of closed subsets of unbounded (not necessarily
separable) metric space $(X, \varrho_{_X})$ endowed with the metric
$\rho_{_X}^{\rm cl}$ introduced in [Zhukovskiy E.S., Panasenko E.A. //
Fixed Point Theory and Applications. 2013:10]. It is shown that if any closed
ball in the space $(X, \varrho_{_X})$ is totaly bounded, then convergence in the space
$\left({\rm clos}(X), \rho_{_X}^{\rm cl}\right)$ of a sequence $\{F_i\}_{i=1}^\infty$ to $F$ is equivalent to convergence in the sense of
Wijsman, that is to convergence for each $x \in X$ of the distances
$\varrho_{_X}(x, F_i)$ to $\varrho_{_X}(x, F).$
Keywords:space of closed subsets of a metric space, Wijsman convergence, metrizability.
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