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Let $(X, d)$ , $(Y, ρ)$ be metric spaces and $f : X \to Y$ an injective mapping. We put ${∥ f ∥}_{Lip} = sup {ρ (f (x), f (y)) / d (x, y); x, y \in X, x \neq y}$ , and $dist (f) = {∥ f ∥}_{Lip} . {∥ f^{- 1} ∥}_{Lip}$ (the distortion of the mapping $f$ ). We investigate the minimum dimension $N$ such that every $n$ -point metric space can be embedded into the space $ℓ_{\infty}^{N}$ with a prescribed distortion $D$ . We obtain that this is possible for $N \geq C {(log n)}^{2} n^{3 / D}$ , where $C$ is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into $ℓ_{p}^{N}$ are obtained by a similar method.

Note on category in Cartesian products of metrizable spaces

Roman Pol (1979)

Fundamenta Mathematicae

Note on commutable functions.

Marek Cezary Zdun (1988)

Aequationes mathematicae

Note on connections of the point of continuity property and Kuratowski problem on function having the Baire property

Ondřej F. K. Kalenda (1997)

Acta Universitatis Carolinae. Mathematica et Physica

Note on countable unions of Corson countably compact spaces

Ondřej F. K. Kalenda (2004)

Commentationes Mathematicae Universitatis Carolinae

We show that a compact space $K$ has a dense set of $G_{δ}$ points if it can be covered by countably many Corson countably compact spaces. If these Corson countably compact spaces may be chosen to be dense in $K$ , then $K$ is even Corson.

In this note we are going to study dense covers in the category of locales. We shall show that any product of finitely regular locales with some dense covering property has this property as well.

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Note on a Compactification Due to Nielsen and Sloyer.

Note on a result of E.P. Lane

Note on a Theorem of Kuratowski-Sierpiński

Note on a theorem of S. Mercourakis about weakly $K$ -analytic Banach spaces

Note on bi-Lipschitz embeddings into normed spaces

Note on category in Cartesian products of metrizable spaces

Note on commutable functions.

Note on connections of the point of continuity property and Kuratowski problem on function having the Baire property

Note on countable unions of Corson countably compact spaces

Note on decompositions of metrizable spaces I

Note on decompositions of metrizable spaces II

Note on Demi-Topological Functors.

Note on dense covers in the category of locales

Note on extremal points

Note on initial topologies on rational vector spaces induced by revalued linear mappings.

Note on KKM maps and applications.

Note on limits of simply continuous and cliquish functions.

Note on metric-dependent dimension functions

Note on metrization of quasi-uniform spaces

Note on quasi-uniform spaces and representable spaces

Currently displaying 141 – 160 of 189