Abstract
The hyperspace consists of all subsets of a vector space. Owing to a lack of additive inverse elements, the hyperspace cannot form a vector space. In this paper, we shall consider a so-called informal norm to the hyperspace in which the axioms regarding the informal norm are almost the same as the axioms of the conventional norm. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets. In this case, the topologies generated by these different concepts of open sets are investigated.
Highlights
IntroductionThe topic of set-valued analysis (or multivalued analysis) has been studied for an extensive period
The topic of set-valued analysis has been studied for an extensive period
The hyperspace denoted by P ( X ) is the collection of all subsets of a vector space X
Summary
The topic of set-valued analysis (or multivalued analysis) has been studied for an extensive period. The set-valued analysis usually studies the mathematical structure in P ( X ) in which each element in P ( X ) is treated as a subset of. We shall endow a so-called informal norm to P ( X ) even though P ( X ) is not a vector space. The main purpose of this paper is to study the topological structures of informally normed space P ( X ). Based on these topological structures, the potential applications in nonlinear analysis, differential inclusion and set-valued optimization (or set optimization) are possible after suitable formulation. Based on the different types of openness, we shall study the topological structure of the normed hyperspace (P ( X ), k · k).
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