Search results for " 54C25"

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Urysohn's metrization theorem for higher cardinals

2011

In this paper a generalization of Urysohn's metrization theorem is given for higher cardinals. Namely, it is shown that a topological space with a basis of cardinality at most $|\omega_\mu|$ or smaller is $\omega_\mu$-metrizable if and only if it is $\omega_\mu$-additive and regular, or, equivalently, $\omega_\mu$-additive, zero-dimensional, and T\textsubscript{0}. Furthermore, all such spaces are shown to be embeddable in a suitable generalization of Hilbert's cube.

Mathematics::Logic54F65 54C25 54A25 54D70 54D10 54D20General Topology (math.GN)FOS: MathematicsMathematics::General TopologyAstrophysics::Cosmology and Extragalactic AstrophysicsMathematics - General Topology
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Isometric embeddings of snowflakes into finite-dimensional Banach spaces

2016

We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.

30L05 46B85 54C25 54E40 28A80Pure mathematicsmetric spacesGeneral MathematicsMathematicsofComputing_GENERALBanach space01 natural sciencesfunctional analysisCardinalityMathematics - Metric GeometryDimension (vector space)0103 physical sciencesFOS: MathematicsMathematics (all)Mathematics::Metric Geometry0101 mathematicsSnowflakeNormed vector spaceMathematicsConcave functionApplied Mathematicsta111010102 general mathematicsnormiavaruudetMetric Geometry (math.MG)normed spacesmetriset avaruudetMetric spacefractalsfraktaalit010307 mathematical physicsfunktionaalianalyysiMathematics (all); Applied MathematicsVector spaceProceedings of the American Mathematical Society
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