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Ramsey, Lebesgue, and Marczewski sets and the Baire property

100%

Reardon P.

Fundamenta Mathematicae

1996

tom 149

nr 3

191-203

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented. THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets. THEOREM. In the Ellentuck topology on $[ω]^ω$, $(s)_0$ is a proper subset of the hereditary ideal associated with (s). We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not $B_r$-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.

On the structure of perfect sets in various topologies associated with tree forcings

63%

Nowik A., Reardon P.

Open Mathematics

2013

tom 11

nr 3

509-518

We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Open Mathematics

2 Reardon P.

1 Nowik A.

1 2013

1 1996

Wyniki wyszukiwania

Ramsey, Lebesgue, and Marczewski sets and the Baire property

On the structure of perfect sets in various topologies associated with tree forcings