A coarse Mayer–Vietoris principle

1993 ◽  
Vol 114 (1) ◽  
pp. 85-97 ◽  
Author(s):  
Nigel Higson ◽  
John Roe ◽  
Guoliang Yu
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Cohomology Theory ◽  
Coarse Geometry ◽  
Locally Compact ◽  
Finite Propagation ◽  
Noncompact Riemannian Manifold

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C*-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

scholarly journals The ideal of Lipschitz classical p-compact operators and its injective hull

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Toufik Tiaiba ◽  
Dahmane Achour
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Operator Ideal ◽  
Linear Case ◽  
Injective Hull ◽  
Nuclear Operators ◽  
The Ideal

Abstract We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

1972 ◽  
Vol 24 (4) ◽  
pp. 622-630 ◽  
Author(s):  
Jack R. Porter ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Hausdorff Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Locally Compact ◽  
Completely Regular ◽  
Isolated Points ◽  
Remote Points ◽  
Topological Boundary ◽  
Regular Closed

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

scholarly journals Compactness Property of Fuzzy Soft Metric Space and Fuzzy Soft Continuous Function

2021 ◽  
pp. 3031-3038
Author(s):  
Raghad I. Sabri
Keyword(s):  
Functional Analysis ◽  
Continuous Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Compactness Property ◽  
Locally Compact ◽  
Fuzzy Metric Spaces ◽  
Fundamental Properties ◽  
Definition Of

      The theories of metric spaces and fuzzy metric spaces are crucial topics in mathematics.    Compactness is one of the most important and fundamental properties that have been widely used in Functional Analysis. In this paper, the definition of compact fuzzy soft metric space is introduced and some of its important theorems are investigated. Also, sequentially compact fuzzy soft metric space and locally compact fuzzy soft metric space are defined and the relationships between them are studied. Moreover, the relationships between each of the previous two concepts and several other known concepts are investigated separately. Besides, the compact fuzzy soft continuous functions are studied and some essential theorems are proved.

scholarly journals Metric completions, the Heine-Borel property, and approachability

2020 ◽  
Vol 18 (1) ◽  
pp. 162-166
Author(s):  
Vladimir Kanovei ◽  
Mikhail G. Katz ◽  
Tahl Nowik
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Short Proof ◽  
Universal Cover

Abstract We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by Do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in ∗M.

Perfect Images of Zero-Dimensional Separable Metric Spaces

1982 ◽  
Vol 25 (1) ◽  
pp. 41-47 ◽  
Author(s):  
Jan Van Mill ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Cantor Set ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Continuous Surjection

AbstractLet Q denote the rationals, P the irrationals, C the Cantor set and L the space C − {p} (where p ∈ C). Let f : X → Y be a perfect continuous surjection. We show: (1) If X ∈ {Q, P, Q × P}, or if f is irreducible and X ∈ {C, L}, then Y is homeomorphic to X if Y is zero-dimensional. (2) If X ∈ {P, C, L} and f is irreducible, then there is a dense subset S of Y such that f|f ← [S] is a homeomorphism onto S. However, if Z is any σ-compact nowhere locally compact metric space then there is a perfect irreducible continuous surjection from Q × C onto Z such that each fibre of the map is homeomorphic to C.

scholarly journals Isometry groups of separable metric spaces

2009 ◽  
Vol 146 (1) ◽  
pp. 67-81 ◽  
Author(s):  
MACIEJ MALICKI ◽  
SŁAWOMIR SOLECKI
Keyword(s):  
Metric Space ◽  
Isometry Group ◽  
Metric Spaces ◽  
Structure Theorem ◽  
Ultrametric Space ◽  
Isometry Groups ◽  
Locally Compact ◽  
Natural Action ◽  
Polish Group ◽  
Separable Metric Space

AbstractWe show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

Decomposition of Metric Spaces with a 0-Dimensional Set of Non-Degenerate Elements

1969 ◽  
Vol 21 ◽  
pp. 202-216 ◽  
Author(s):  
Jack W. Lamoreaux
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Open Covering ◽  
Locally Compact ◽  
Locally Finite ◽  
Decomposition Space ◽  
Dimensional Decomposition ◽  
C Decomposition ◽  
Condition B

Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.

scholarly journals Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem

2019 ◽  
Vol 20 (5) ◽  
pp. 1035-1133
Author(s):  
Charles Fefferman ◽  
Sergei Ivanov ◽  
Yaroslav Kurylev ◽  
Matti Lassas ◽  
Hariharan Narayanan
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Injectivity Radius ◽  
Principal Curvatures ◽  
Curvature Bounds ◽  
Fixed Dimension ◽  
Manifold Reconstruction

Abstract We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space $$(X,d_X)$$ ( X , d X ) . This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold $$S\subset {{\mathbb {R}}}^m$$ S ⊂ R m , $$m>n$$ m > n needs to be constructed to approximate a point cloud in $${{\mathbb {R}}}^m$$ R m . These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov–Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in $${{\mathbb {R}}}^m$$ R m and interpolated to a smooth submanifold.

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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