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A topological space is called connected if it is not the union of two disjoint, nonempty and open sets in this space. The standard exercises show that here the concept of open sets can be replaced by closed sets or separated sets. In this context we will discuss the definition of connected sets in topological spaces, not being the whole space with particular regard to metric spaces, without the term...
In this paper we prove that if the composition operator H of generator h : Iba× C → Y (X is a real normed space, Y is a real Banach space, C is a convex cone in X and Iba⸦ R2) maps Φ1BV (Iba, C) into Φ2BV (Iba, Y) and is uniformly bounded, then the left-left regularization h* of h is an affine function in the third variable.
We prove in this paper that if the composition operator H, generated by a function h : Ibax C(Iba) Y , maps ɸBV (Iba ,C) into ɸ2 BV (Iba , Y ) and is uniformly continuous, then the left-left regularization h* of h is an affine function with respect to the third variable.
We prove that, under some general assumptions, the one-sided regularizations of the generator of any uniformly bounded set-valued composition operator, acting in the spaces of functions of bounded variation in the sense of Schramm with nonempty bounded closed and convex values is an affine function. As a special case, we obtain an earlier result ([15]).
We show that the one-sided regularizations of the generator of any uniformly continuous set-valued Nemytskij operator, acting between the spaces of functions of bounded variation in the sense of Schramm, is an affine function. Results along these lines extend the study [1].
It is known that every locally defined operator acting between two Hölder spaces is a Nemytskii superposition operator. We show that if such an operator is bounded in the sense of the norm, then its generator is continuous.
A generalization theorem for i-connected sets in the Hashimoto topology is given. Moreover, i-connectivity in the topology of at most countable complements the order topology is presented.
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