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Ramsey, Lebesgue, and Marczewski sets and the Baire property

Patrick Reardon — 1996

Fundamenta Mathematicae

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...

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

Andrzej Nowik;  Patrick Reardon — 2013

Open Mathematics

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.

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