Unions of rectifiable curves in Euclidean space and the covering number of the meagre ideal

1999 ◽  
Vol 64 (2) ◽  
pp. 701-726 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Euclidean Space ◽  
Banach Spaces ◽  
Set Theory ◽  
Metric Space ◽  
Covering Number ◽  
Cardinal Invariant ◽  
Rectifiable Curves ◽  
Straight Lines

AbstractTo any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ1 is meagre yet there are ℵ1 rectifiable curves in ℝ3 whose union is not meagre. The consistency of this statement when the phrase “rectifiable curves” is replaced by “straight lines” remains open.

On a geometric statement of Ramsey type

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Set Theory ◽  
Ramsey Type ◽  
Straight Lines

Abstract A simple geometric assertion of Ramsey type, concerning families of straight lines in the Euclidean space R 3 \mathbb{R}^{3} , is formulated, and it is shown that the assertion turns out to be undecidable within the framework of ZFC set theory.

scholarly journals Tight Embeddability of Proper and Stable Metric Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
F. Baudier ◽  
G. Lancien
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Space ◽  
Metric Spaces ◽  
Proper Subset ◽  
Reflexive Banach Spaces

Abstract We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [1,∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the ℓpn’s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.

scholarly journals ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

2013 ◽  
Vol 56 (3) ◽  
pp. 519-535 ◽  
Author(s):  
TIMOTHY FAVER ◽  
KATELYNN KOCHALSKI ◽  
MATHAV KISHORE MURUGAN ◽  
HEIDI VERHEGGEN ◽  
ELIZABETH WESSON ◽  
...  
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Independent Set ◽  
Ultrametric Space ◽  
Ultrametric Spaces ◽  
Negative Type ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Generalized Roundness

AbstractMotivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

MAXIMAL COMPUTABILITY STRUCTURES

2016 ◽  
Vol 22 (4) ◽  
pp. 445-468 ◽  
Author(s):  
ZVONKO ILJAZOVIĆ ◽  
LUCIJA VALIDŽIĆ
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Dense Sequence

AbstractA computability structure on a metric space is a set of sequences which satisfy certain conditions. Of a particular interest are those computability structures which contain a dense sequence, so called separable computability structures. In this paper we observe maximal computability structures which are more general than separable computability structures and we examine their properties. In particular, we examine maximal computability structures on subspaces of Euclidean space, we give their characterization and we investigate conditions under which a maximal computability structure on such a space is unique. We also give a characterization of separable computability structures on a segment.

scholarly journals On Constant Metric Dimension of Some Generalized Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xuewu Zuo ◽  
Abid Ali ◽  
Gohar Ali ◽  
Muhammad Kamran Siddiqui ◽  
Muhammad Tariq Rahim ◽  
...  
Keyword(s):  
Image Processing ◽  
Euclidean Space ◽  
Global Positioning System ◽  
Metric Space ◽  
Network Theory ◽  
Convex Polytopes ◽  
Metric Dimension ◽  
Positioning System ◽  
The Family ◽  
Global Positioning

Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) to the arbitrary metric space. A family ℱ = G n of connected graphs with n ≥ 3 is a family of constant metric dimension if dim G = k (some constant) for all graphs in the family. Family ℱ has bounded metric dimension if dim G n ≤ M , for all graphs in ℱ . Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.

scholarly journals On constructing completions

2005 ◽  
Vol 70 (3) ◽  
pp. 969-978 ◽  
Author(s):  
Laura Crosilla ◽  
Hajime Ishihara ◽  
Peter Schuster
Keyword(s):  
Set Theory ◽  
Metric Space ◽  
Ordered Set ◽  
Proper Class ◽  
Original Space ◽  
Full Form ◽  
Dedekind Cuts ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

AbstractThe Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo–Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two–element coverings is used.In particular, the Dedekind reals form a set: whence we have also refined an earlier result by Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class: in particular, every complete separable metric space automatically is a set.

Linear Functions with Two Points of Intersection?

1995 ◽  
Vol 88 (5) ◽  
pp. 376-378
Author(s):  
Michael A. Contino
Keyword(s):  
Euclidean Space ◽  
Graphing Calculator ◽  
Linear Functions ◽  
First Year ◽  
Systems Of Equations ◽  
College Mathematics ◽  
Algebra Students ◽  
Straight Lines

Of course two straight lines in Euclidean space cannot intersect in more than one point unless they are the same line and intersect everywhere—or can they? Follow this problem on the graphing calculator, however, and the surprising twist that gives this article its name will be seen. The material covered should be readily accessible to first-year-algebra students who have studied systems of equations, but it also contains valuable lessons for college mathematics professors who have been easily deceived by its apparent simplicity and familiarity.

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