scholarly journals Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

2019 ◽  
Vol 3 (1) ◽  
pp. 1-28
Author(s):  
Christian Ronse ◽  
Loic Mazo ◽  
Mohamed Tajine
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Hausdorff Distance ◽  
Closed Subset ◽  
Minimum Distance ◽  
Topological Properties ◽  
Closed Set ◽  
Adjacency Graph ◽  
Discrete Subspace ◽  
Partial Connection

Abstract We consider Hausdorff discretization from a metric space E to a discrete subspace D, which associates to a closed subset F of E any subset S of D minimizing the Hausdorff distance between F and S; this minimum distance, called the Hausdorff radius of F and written rH(F), is bounded by the resolution of D. We call a closed set F separated if it can be partitioned into two non-empty closed subsets F1 and F2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in ℝn and ℤn), we show that given a non-separated closed subset F of E, for any r > rH(F), every Hausdorff discretization of F is connected for the graph with edges linking pairs of points of D at distance at most 2r. When F is connected, this holds for r = rH(F), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on D of the balls of radius rH(F). However, when the closed set F is separated, the Hausdorff discretizations are disconnected whenever the resolution of D is small enough. In the particular case where E = ℝn and D = ℤn with norm-based distances, we generalize our previous results for n = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called coordinate-homogeneous norms, which include the Lp norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.

scholarly journals On the Nielsen root theory of n-valued maps

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Robert F. Brown
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Lie Group ◽  
Hausdorff Distance ◽  
Root Number ◽  
Root Numbers ◽  
Fixed Point Result

AbstractLet $$\phi :X \multimap Y$$ ϕ : X ⊸ Y be an n-valued map of connected finite polyhedra and let $$a \in Y$$ a ∈ Y . Then, $$x \in X$$ x ∈ X is a root of $$\phi $$ ϕ at a if $$a \in \phi (x)$$ a ∈ ϕ ( x ) . The Nielsen root number $$N(\phi : a)$$ N ( ϕ : a ) is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$ ϕ . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$ ϵ > 0 , there is an n-valued map $$\psi $$ ψ homotopic to $$\phi $$ ϕ within Hausdorff distance $$\epsilon $$ ϵ of $$\phi $$ ϕ such that $$\psi $$ ψ has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$ q ≠ 2 , then there is an n-valued map homotopic to $$\phi $$ ϕ that has $$N(\phi : a)$$ N ( ϕ : a ) roots at a. We verify the conjecture when $$X = Y$$ X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.

On topological properties of weakly m-convex sets

2021 ◽  
Vol 34 ◽  
pp. 75-84
Author(s):  
Tetiana Osipchuk
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Topological Classification ◽  
Connected Components ◽  
Topological Properties ◽  
Convex Domains ◽  
Closed Set ◽  
Dimensional Plane ◽  
Open Set ◽  
Plane Passing

The topological properties of classes of generally convex sets in multidimensional real Euclidean space $\mathbb{R}^n$, $n\ge 2$, known as $m$-convex and weakly $m$-convex, $1\le m<n$, are studied in the present work. A set of the space $\mathbb{R}^n$ is called \textbf{\emph{$m$-convex}} if for any point of the complement of the set to the whole space there is an $m$-dimensional plane passing through this point and not intersecting the set. An open set of the space is called \textbf{\emph{weakly $m$-convex}}, if for any point of the boundary of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of the space is called \textbf{\emph{weakly $m$-convex}} if it is approximated from the outside by a family of open weakly $m$-convex sets. These notions were proposed by Professor Yuri Zelinskii. It is known the topological classification of (weakly) $(n-1)$-convex sets in the space $\mathbb{R}^n$ with smooth boundary. Each such a set is convex, or consists of no more than two unbounded connected components, or is given by the Cartesian product $E^1\times \mathbb{R}^{n-1}$, where $E^1$ is a subset of $\mathbb{R}$. Any open $m$-convex set is obviously weakly $m$-convex. The opposite statement is wrong in general. It is established that there exist open sets in $\mathbb{R}^n$ that are weakly $(n-1)$-convex but not $(n-1)$-convex, and that such sets consist of not less than three connected components. The main results of the work are two theorems. The first of them establishes the fact that for compact weakly $(n-1)$-convex and not $(n-1)$-convex sets in the space $\mathbb{R}^n$, the same lower bound for the number of their connected components is true as in the case of open sets. In particular, the examples of open and closed weakly $(n-1)$-convex and not $(n-1)$-convex sets with three and more connected components are constructed for this purpose. And it is also proved that any compact weakly $m$-convex and not $m$-convex set of the space $\mathbb{R}^n$, $n\ge 2$, $1\le m<n$, can be approximated from the outside by a family of open weakly $m$-convex and not $m$-convex sets with the same number of connected components as the closed set has. The second theorem establishes the existence of weakly $m$-convex and not $m$-convex domains, $1\le m<n-1$, $n\ge 3$, in the spaces $\mathbb{R}^n$. First, examples of weakly $1$-convex and not $1$-convex domains $E^p\subset\mathbb{R}^p$ for any $p\ge3$, are constructed. Then, it is proved that the domain $E^p\times\mathbb{R}^{m-1}\subset\mathbb{R}^n$, $n\ge 3$, $1\le m<n-1$, is weakly $m$-convex and not $m$-convex.

A Maxmin Distance Problem

1972 ◽  
Vol 94 (2) ◽  
pp. 155-158 ◽  
Author(s):  
R. Aggarwal ◽  
G. Leitmann
Keyword(s):  
Minimum Distance ◽  
Terminal State ◽  
Initial State ◽  
Closed Set ◽  
Distance Problem

The problem of maximizing the minimum distance of a dynamical system’s state from a given closed set, while transferring the system from a given initial state to a given terminal state, is considered. Two different methods of solution of this problem are given.

Intuitionistic Fuzzy 2-Metric Space and Some Topological Properties

2013 ◽  
pp. 245-260
Author(s):  
Q.M. Danish Lohani
Keyword(s):  
Metric Space ◽  
Normed Space ◽  
Fuzzy Metric Space ◽  
Topological Properties ◽  
Fuzzy Normed Space ◽  
Fuzzy Metric ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Fuzzy Normed Space ◽  
Norm Space

The notion of intuitionistic fuzzy metric space was introduced by Park (2004) and the concept of intuitionistic fuzzy normed space by Saadati and Park (2006). Recently Mursaleen and Lohani introduced the concept of intuitionistic fuzzy 2-metric space (2009) and intuitionistic fuzzy 2-norm space. This paper studies precompactness and metrizability in this new setup of intuitionistic fuzzy 2-metric space.

More on Extending Continuous Pseudometrics

1970 ◽  
Vol 22 (5) ◽  
pp. 984-993 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closed Subset ◽  
Locally Finite ◽  
Whole Space

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

scholarly journals Super-extremal processes and the argmax process

1994 ◽  
Vol 31 (04) ◽  
pp. 958-978 ◽  
Author(s):  
Sidney I. Resnick ◽  
Rishin Roy
Keyword(s):  
Metric Space ◽  
Semicontinuous Function ◽  
Closed Set ◽  
Continuous Choice ◽  
Extremal Processes ◽  
Upper Semicontinuous Function ◽  
Classical Vector ◽  
Vector Valued ◽  
Path Properties ◽  
Separable Metric Space

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processes Y = {Yt, t &gt; 0}. At any t &gt; 0, Y t is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t &gt; 0, Y t is associated. For a compact metric E, we consider the argmax process M = {Mt, t &gt; 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.

scholarly journals On convergence of closed sets in a metric space and distance functions

1985 ◽  
Vol 31 (3) ◽  
pp. 421-432 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Distance Functions ◽  
Convergence Theorems ◽  
Closed Set ◽  
Closed Sets ◽  
The Family ◽  
Kuratowski Convergence ◽  
General Convergence

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

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