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A new graph topology. Connections with the compact open topology. (English) Zbl 0836.54010
Let \(E\) be a closed connected subset of \(\mathbb{R}\) and let \({\mathcal C}\) be the set of all the closed non-empty subsets of \(E\). Given \(\Omega \in {\mathcal C}\), let \(G_\Omega\) denote the set of all the graphs of continuous functions in \(C (\Omega, \mathbb{R}^m)\). Let \(G = \bigcup_{\Omega \in {\mathcal C}} G_\Omega\). We endow \(G\) with a new topology called \(\tau\)- topology. It is strictly coarser than the Hausdorff metric topology and strictly finer than the topology of uniform convergence of distance functionals on bounded sets of \(\mathbb{R}^{m + 1}\). The topological space \((G, \tau)\) is homeomorphic to the quotient space \([({\mathcal C}, \tau) \times C (E, \mathbb{R}^m)]/{\mathcal R}\) with a suitable equivalence relation \({\mathcal R}\). The homeomorphic property of the \(\tau\)-topology has a great relevance in the study of functional differential equations. In fact, it allows us to prove existence, uniqueness and continuous dependence of the solutions of hereditary differential equations by means of classical fixed point theorems applied to the homeomorphic functional space. For what concerns the relationships between \(\tau\)-topology and the topologies introduced in \(G_\Omega\) by other authors, note that none of the last ones makes \(G_\Omega\) homeomorphic to the topological space \(C (\Omega, \mathbb{R}^m)\) for arbitrary \(\Omega \in {\mathcal C}\), unless we assume regularity assumption on \(\Omega\). But any regularity assumption on \(\Omega\) would compromise the generality of hereditary structure on differential equations.

54C35 Function spaces in general topology
34K05 General theory of functional-differential equations
54C20 Extension of maps
Full Text: DOI
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