Classical measure spaces in the Ellentuck and density topologies.

摘要

It is shown that in the Ellentuck topology T every T-dense set contains a countable T-perfect set and that there is no T-Bernstein subdivision of a countable T-perfect set. Examples of sets which are both L{dollar}sb0{dollar} and T-first category but not T-B{dollar}sb{lcub}rm r{rcub}{dollar} are constructed. One of these examples is constructed using a Vitali-type construction in ({dollar}omega{dollar}). The other is a Euclidean-Bernstein subset of a particular T-perfect set. In addition, an example of a set which is in the hereditary ideal associated with T-(s) but which does not belong to T-(s){dollar}sb0{dollar} is exhibited.; Oxtoby (O, p.89) has shown that in the density topology D, the {dollar}sigma{dollar}-algebras D-B{dollar}sb{lcub}rm w{rcub}{dollar} and L coincide. Scheinberg (Sc) has shown that even D-Borel = L. It is shown that D-(s) = L. Further it is shown that for any topology {dollar}Im{dollar} which is regular and in which {dollar}Im{dollar}-first category sets are {dollar}Im{dollar}-scattered, {dollar}Im{dollar}-B{dollar}sb{lcub}rm w{rcub}{dollar} = {dollar}Im{dollar}-(s). In addition, it is shown that the hereditary ideals associated with D-(s) and CR equal D-(s){dollar}sb0{dollar} and CR{dollar}sb0{dollar}, respectively.; It is shown that the completely Ramsey-measureable sets satisfy the hypothesis of the so-called "Hull Theorem" of Marczewski and from this a relatively simple proof that all analytic sets are Ramsey is obtained. Finally, a question posed by B. Anizcyck (A) is answered concerning the union of (s){dollar}sb0{dollar} sets.