scholarly journals Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

2021 ◽  
Author(s):  
Ramón Aliaga ◽  
Chris Gartland ◽  
Colin Petitjean ◽  
Antonín Procházka
Keyword(s):  
Metric Spaces ◽  
Lipschitz Functions

scholarly journals Two classes of metric spaces

2016 ◽  
Vol 17 (1) ◽  
pp. 57 ◽  
Author(s):  
Isabel Garrido ◽  
Ana S. Meroño
Keyword(s):  
Metric Spaces ◽  
Lipschitz Functions ◽  
The Other ◽  
Metric Properties ◽  
Other Hand ◽  
Common Framework

<p>The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.</p>

scholarly journals Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces

2000 ◽  
Vol 103 (2) ◽  
pp. 292-301 ◽  
Author(s):  
Salvador Romaguera ◽  
Manuel Sanchis
Keyword(s):  
Best Approximation ◽  
Metric Spaces ◽  
Lipschitz Functions

scholarly journals Pointwise Lipschitz functions on metric spaces

2010 ◽  
Vol 363 (2) ◽  
pp. 525-548 ◽  
Author(s):  
E. Durand-Cartagena ◽  
J.A. Jaramillo
Keyword(s):  
Metric Spaces ◽  
Lipschitz Functions

Point derivations and cohomologies of Lipschitz algebras

2019 ◽  
Vol 62 (4) ◽  
pp. 1173-1187
Author(s):  
Kazuhiro Kawamura
Keyword(s):  
Metric Spaces ◽  
Banach Algebras ◽  
Studia Math ◽  
Lipschitz Functions ◽  
Homological Dimension ◽  
Smooth Functions ◽  
Infinite Dimensional ◽  
Higher Dimensional ◽  
Geodesic Metric ◽  
Dual Module

AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

scholarly journals CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS

2004 ◽  
Vol 04 (02) ◽  
pp. 109-145 ◽  
Author(s):  
STEFAN GESCHKE ◽  
MARTIN GOLDSTERN ◽  
MENACHEM KOJMAN
Keyword(s):  
Set Theory ◽  
Metric Spaces ◽  
Ramsey Theory ◽  
Polish Space ◽  
Lipschitz Functions ◽  
Cantor Space ◽  
Polish Spaces ◽  
Real Functions ◽  
Number Formula ◽  
Independence Results

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]2→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2ω, c min and c max , which satisfy [Formula: see text] and prove: Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X]2→2with[Formula: see text], [Formula: see text] (2) There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of (2ω)2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which (1) ℝ2 is coverable byℵ1graphs and reflections of graphs of continuous real functions; (2) ℝ2 is not coverable byℵ1graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row (3).

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