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**The (largest) Lebesgue number and its relative version(****arXiv****)**

**Author : **Vera Tonić

**Abstract : **In this paper we compare different definitions of the (largest) Lebesgue number of a cover U for a metric space X. We also introduce the relative version for the Lebesgue number of a covering family U for a subset A⊆X, and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from S. Buyalo — N. Lebedeva paper “Dimensions of locally and asymptotically self-similar spaces”, involving λ-quasi homothetic maps with coefficient R between metric spaces, and the comparison of the mesh and the Lebesgue number of a covering family for a subset on both sides of the map.

**2. Lebesgue Number and Total Boundedness(****arXiv****)**

**Author :** Ajit Kumar Gupta, Saikat Mukherjee

**Abstract : **A generalization of the Lebesgue number lemma is obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space X has a Lebesgue number, then X is totally bounded. A property of metric spaces which is a generalization of connectedness and Menger convexity is introduced. It is observed that Atsujiness and compactness are equivalent for a metric space with this introduced property as well as for a chainable metric space

**3. Exponential Decay of Lebesgue Numbers(****arXiv****)**

**Author : **Peng Sun

**Abstract : **We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.

**4.Lebesgue numbers and Atsuji spaces in subsystems of second order arithmetic(****arXiv****)**

**Author :** Mariagnese Giusto, Alberto Marcone

**Abstract :**We study properties of complete separable metric spaces within the framework of subsystems of second order arithmetic. In particular we consider Lebesgue and Atsuji spaces. The former are those such that every open covering U has a Lebesgue number, i.e. a positive number q such that for every point x of the space, there exists an element of U which contains the ball of center x and radius q; the latter are those such that every continuous function into another complete separable metric space is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to WKL_0; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to ACA_0; the statement “every Lebesgue space is Atsuji” is provable in RCA_0; the statement “every Atsuji space is Lebesgue” is provable in ACA_0, but we do not know if it is equivalent to ACA_0. We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to Pi¹_1-CA_0; the statements “there exists a complete separable metric space which is perfect and Heine-Borel compact (resp. Lebesgue, Atsuji)” are all equivalent to WKL_0.

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