Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5777743 | Topology and its Applications | 2017 | 20 Pages |
Abstract
Let (X,d) be a complete, bounded, metric space. For a nonempty, closed subset A of X denote by Câ(AÃA) the set of all continuous, bounded, real-valued functions on AÃA. Denote byCâ =â{Câ(AÃA)|A is a nonempty closed subset of X} the set of all partial, continuous and bounded functions. We prove that there exists a linear, regular extension operator from Câ endowed with the topology of convergence in the Hausdorff distance of graphs of partial functions to the space Câ(XÃX) with the topology of uniform convergence on compact sets. The constructed extension operator preserves constant functions, pseudometrics, metrics and admissible metrics. For a fixed, nonempty, closed subset A of X the restricted extension operator from Câ(AÃA) to Câ(XÃX) is continuous with respect to the topologies of pointwise convergence, uniform convergence on compact sets and uniform convergence considered on both Câ(AÃA) and Câ(XÃX).
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Geometry and Topology
Authors
T. Banakh, I. Stasyuk, E.D. Tymchatyn, M. Zarichnyi,