A hypertopology intended for functional differential equations

Abstract

Let (X,d) be a locally compact separable metric space, Y a Fréchet space and let C be the of all the closed non-empty susbsets of X. Given ω ∈ C let G _ω denote the set of all the graphs of continuous functions in C(ω Y). Let G=u_ω∈cG_ωWe endow G with a new topology called τ-topology. The topological space (G,τ) is homeomorphic to the quotient space [(C,r)×C(X,Y)]/R with respect to a suitable equivalence relation R. The relationships between τ-topology and the topologies in C_ω by other Authors are explored. The results here obtained generlize those got in [12] and find applications in the theory of Ordinary and Partial Differential Equations with hereditary structure.

Keywords:

• graph topology compact-open topology
• functional differential equations