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Opérations de Hausdorff itérées et réunions croissantes de compacts

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Kahane S.

Fundamenta Mathematicae

1992

tom 141

nr 2

169-194

In this paper, motivated by questions in Harmonic Analysis, we study the operation of (countable) increasing union, and show it is not idempotent: $ω_1$ iterations are needed in general to obtain the closure of a class under this operation. Increasing union is a particular Hausdorff operation, and we present the combinatorial tools which allow to study the power of various Hausdorff operations, and of their iterates. Besides countable increasing union, we study in detail a related Hausdorff operation, which preserves compactness.

Finite union of H-sets and countable compact sets

100%

Kahane S.

Colloquium Mathematicum

1993

tom 65

nr 1

In [2], D. E. Grow and M. Insall construct a countable compact set which is not the union of two H-sets. We make precise this result in two directions, proving such a set may be, but need not be, a finite union of H-sets. Descriptive set theory tools like Cantor-Bendixson ranks are used; they are developed in the book of A. S. Kechris and A. Louveau [6]. Two proofs are presented; the first one is elementary while the second one is more general and useful. Using the last one I prove in my thesis, directed by A. Louveau, the existence of a countable compact set which is not a finite union of Dirichlet sets. This result, quoted in [3], is weaker because all Dirichlet sets belong to H. Other new results about the class H and similar classes of thin sets can be found in [4], [1] and [5].

Ograniczanie wyników

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

2 Kahane S.

1 1993

1 1992

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Opérations de Hausdorff itérées et réunions croissantes de compacts

Finite union of H-sets and countable compact sets