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A new graph topology intended for functional differential equations. (English) Zbl 0890.54010

Summary: Let \((X,d)\) be a boundedly compact metric space and let \(\mathcal C\) be the set of all the closed non-empty subsets of \(X\). 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. The topological space \((G,\tau)\) is homeomorphic to the quotient space \([({\mathcal C},\tau)\times C(X,\mathbb{R}^m)]/{\mathcal R}\) with a suitable equivalence relation \(\mathcal R\). The relationships between \(\tau\)-topology and the topologies introduced in \(G_\Omega\) by other authors are explored. The obtained results generalize the findings of our paper [Appl. Anal. 53, No. 3-4, 185-196 (1994; Zbl 0836.54010)] and can be applied in the theory of ordinary and partial differential equations with hereditary structure.

MSC:

54B20 Hyperspaces in general topology
34K05 General theory of functional-differential equations
35R10 Partial functional-differential equations
54C30 Real-valued functions in general topology
54B15 Quotient spaces, decompositions in general topology
54C20 Extension of maps
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)

Citations:

Zbl 0836.54010
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