![]() We also include a study of path connectedness, particularly we prove that for a smooth dendroid this hyperspace is pathwise connected, and we present a general result which implies that for an Euclidean space this hyperspace has uncountably many arc components.Ģ010 Mathematics Subject Classification. We show results of compactness, connectedness and local connectedness for this hyperspace. 04510, MéxicoĮ-mail: de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla,Į-mail: rompc this paper we study the hyperspace of all nonempty closed totally disconnected subsets of a space, equipped with the Vietoris topology. 72570, Puebla, MéxicoĮ-mail: de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Ciudad de México, C.P. San Manuel, Edificio FM3-210,Ĭiudad Universitaria C.P. THE HYPERSPACE OF TOTALLY DISCONNECTED SETS Raúl Escobedo, Patricia Pellicer-Covarrubias and Vicente Sánchez-Gutiérrezįacultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla,Īv. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.The hyperspace of totally disconnected sets Glasnik Matematicki, Vol. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. ![]() Also, there are idempotent counterparts of the convex sets these include the so-called max-plus and max min convex sets. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. ![]() The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In recent decades, we have seen intensive research in this direction. Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. We do not consider here the uniqueness of a convex subset with given boundary data. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$. Specifically, we consider the data composed of the conformal structure on the boundary of $K$ in the Poincar\'e model of $\HH^3$, together with the induced metric on $\partial K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. ![]() It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>-1$ can be obtained. The induced metric on $\partial K$ then has curvature $K>-1$. Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. For every tuple $d_1,\dots, d_l\geq 2,$ let $\mathbb)$, with $T$ a positive definite linear isomorphism of $\mathbb R^n$.
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