We show that a completely regular space Y is a p-space (a Čech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.
Following the paper [BDC1], further relations between the classical topologies on function spaces and new ones induced by hyperspace topologies on graphs of functions are introduced and further characterizations of boundedly spaces are given.
The object of this paper is the Hausdorff metric topology on partial maps with closed domains. This topological space is denoted by . An equivalence of well-posedness of constrained continuous problems is proved. By using the completeness of the Hausdorff metric on the space of usco maps with moving domains, the complete metrizability of is investigated.
This paper completes and improves results of [10]. Let , be two metric spaces and be the space of all -valued continuous functions whose domain is a closed subset of . If is a locally compact metric space, then the Kuratowski convergence and the Kuratowski convergence on compacta coincide on . Thus if and are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology (generated by the box metric of and ) and convergence on ,...
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