We apply the Gromov–Hausdorff metric $$d_G$$ for characterization of certain generalized manifolds. Previously, we have proven that with respect to $$d_G,$$ n-manifolds are limits spaces which obtained by gluing two topological a controlled homotopy equivalence (the so-called 2-patch spaces). In present paper, consider manifold-like $$X^{n},$$ introduced in 1966 Mardeić and Segal, characterized...

It is possible for one to define fractals as those sets which have non-integral Hausdorff Dimension. This paper defines Hausdorff Dimension and rigorously introduces the necessary theory to prove that one such set is indeed a fractal. The first chapter includes a proof of Banach’s Fixed Point Theorem. The Hausdorff Metric is defined and it is mentioned how one produces ‘generators’ for creating...

Let B = {b1, . . . , bK} denote a set of pure outcomes. Let ∆(B) denote the set of probability distributions on B. Finally, let denote a preference relation on the set of nonempty subsets of ∆(B) where this space is endowed with the Hausdorff topology. Let dh(x, y) denote the Hausdorff distance between x and y. ∗Economics Dept., Northwestern University, and School of Economics, Tel Aviv Univers...

Hyperspace theory has its beginnings in the early years of XX century with the work of Felix Hausdorff (1868-1942) and Leopold Vietoris (1891-2002). Given a topological space X, the hyperspace 2X of all nonempty closed subsets of X is equipped with the Vietoris topology, also called the exponential topology, see [37, p. 160] or the finite topology, see [48, p. 153], introduced in 1922 by Vietor...

The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space. We start our investigation from the euclidean case by proving that if a positive Ra...

We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley’s theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω1, then it has a unique complete model of density character λ for every λ ≥ ω1.

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [1]. If M is a non degenerate multifractal measure with associated metric ρ(x, y) = M([x, y]) and structure function ζ , we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimρH with respect to ρ...

We study random recursive constructions with finite “memory” in complete metric spaces and the Hausdorff dimension of the generated random fractals. With each such construction and any positive number β we associate a linear operator V (β) in a finite dimensional space. We prove that under some conditions on the random construction the Hausdorff dimension of the fractal coincides with the value...

We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work cla...