Stability of the ball-covering property

This chapter begins with a discussion of the law on theft, robbery, assault with intent to rob, handling stolen goods, and money laundering offences. The second part of the chapter focuses on the theory of theft, covering property offences; the debate over Gomez; the Hinks debate; temporary appropriation; dishonesty; robberies; and handling stolen goods.

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Small Coverings with Smooth Functions under the Covering Property Axiom

Canadian Journal of Mathematics ◽

10.4153/cjm-2005-020-8 ◽

2005 ◽

Vol 57 (3) ◽

pp. 471-493 ◽

Cited By ~ 5

Author(s):

Krzysztof Ciesielski ◽

Janusz Pawlikowski

Keyword(s):

Borel Function ◽

Covering Property ◽

Smooth Functions ◽

Differentiable Functions ◽

Perfect Set ◽

Covering Property Axiom

AbstractIn the paper we formulate a Covering Property Axiom, CPAprism, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.(a) There exists a family ℱ of less than continuummany functions from ℝ to ℝ such that ℝ2 is covered by functions from ℱ, in the sense that for every 〈x, y〉 ∈ ℝ2 there exists an f ∈ ℱ such that either f (x) = y or f (y) = x.(b) For every Borel function f : ℝ → ℝ there exists a family ℱ of less than continuum many “” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of f.(c) For every n > 0 and a Dn function f: ℝ → ℝ there exists a family ℱ of less than continuum many Cn functions whose graphs cover the graph of f.We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiĭ.

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Some Properties of Hyperspaces with Applications to Continua Theory

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-021-9 ◽

1979 ◽

Vol 31 (1) ◽

pp. 197-210 ◽

Cited By ~ 5

Author(s):

J. Grispolakis ◽

Sam B. Nadler ◽

E. D. Tymchatyn

Keyword(s):

Covering Property ◽

University Of Houston ◽

Chainable Continua ◽

The University

In 1972, Lelek introduced the notion of Class (W) in his seminar at the University of Houston [see below for definitions of concepts mentioned here]. Since then there has been much interest in classifying and characterizing continua in Class (W). For example, Cook has a result [5, Theorem 4] which implies that any hereditarily indecomposible continuum is in Class (W) Read [21, Theorem 4] showed that all chainable continua are in Class (W), and Feuerbacher proved the following result:(1.1) THEOREM [7, Theorem 7]. A non-chainable circle-like continuum is in Class (W) if and only if it is not weakly chainableIn [14, 4.2 and section 6], a covering property (denoted here and in [18] by CP) was defined and studied primarily for the purpose of proving that indecomposability is a Whitney property for the class of chainable continua [14, 4.3].

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Ball-covering property of Banach spaces

Israel Journal of Mathematics ◽

10.1007/bf02773827 ◽

2006 ◽

Vol 156 (1) ◽

pp. 111-123 ◽

Cited By ~ 13

Author(s):

Lixin Cheng

Keyword(s):

Banach Spaces ◽

Covering Property

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Besicovitch Covering Property on graded groups and applications to measure differentiation

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0051 ◽

2019 ◽

Vol 2019 (750) ◽

pp. 241-297 ◽

Cited By ~ 2

Author(s):

Enrico Le Donne ◽

Séverine Rigot

Keyword(s):

Riemannian Manifold ◽

Borel Measure ◽

Blow Up ◽

General Study ◽

Covering Property ◽

Locally Finite ◽

Homogeneous Groups ◽

Finite Borel Measure ◽

Stratified Group ◽

Differentiation Theorem

Abstract We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.

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