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Interpolation of κ -compactness and PCF

István Juhász, Zoltán Szentmiklóssy (2009)

Commentationes Mathematicae Universitatis Carolinae

We call a topological space κ -compact if every subset of size κ has a complete accumulation point in it. Let Φ ( μ , κ , λ ) denote the following statement: μ < κ < λ = cf ( λ ) and there is { S ξ : ξ < λ } [ κ ] μ such that | { ξ : | S ξ A | = μ } | < λ whenever A [ κ ] < κ . We show that if Φ ( μ , κ , λ ) holds and the space X is both μ -compact and λ -compact then X is κ -compact as well. Moreover, from PCF theory we deduce Φ ( cf ( κ ) , κ , κ + ) for every singular cardinal κ . As a corollary we get that a linearly Lindelöf and ω -compact space is uncountably compact, that is κ -compact for all uncountable cardinals κ .

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

Let ( M , d ) be a bounded countable metric space and c > 0 a constant, such that d ( x , y ) + d ( y , z ) - d ( x , z ) c , for any pairwise distinct points x , y , z of M . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of .

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