Displaying similar documents to “Note on certain s-dimensional sets”

Hausdorff measures and two point set extensions

Jan Dijkstra, Kenneth Kunen, Jan van Mill (1998)

Fundamenta Mathematicae

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We investigate the following question: under which conditions is a σ-compact partial two point set contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a sufficient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any σ-compact partial two point set such that its square has Hausdorff 1-measure zero is extendable.

Hausdorff measures and the Morse-Sard theorem.

Carlos Gustavo T. de A. Moreira (2001)

Publicacions Matemàtiques

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Let F : U ⊂ R → R be a differentiable function and p < m an integer. If k ≥ 1 is an integer, α ∈ [0, 1] and F ∈ C, if we set C(F) = {x ∈ U | rank(Df(x)) ≤ p} then the Hausdorff measure of dimension (p + (n-p)/(k+α)) of F(C(F)) is zero.