Regularity properties on the generalized reals

Questions on generalised Baire spaces

“…In this setting, the Baire property is the same as ‐measurability for being the κ‐Cohen forcing on 2 κ . A first systematic study of such regularity properties, where was a suitably generalised version of Cohen, Sacks, Miller, Laver, Mathias and Silver forcing, was conducted in , where it was established that (1) all Borel sets satisfy ‐measurability for all ; (2) sets do not satisfy ‐measurability for any , and (3) ‐measurability for sets is independent, and the implications between statements of the form “all sets are ‐measurable” follows the pattern shown in Figure , in parallel to the situation on the level in the classical setting.…”

“…Recall the regularity properties generalising the Baire property from Definition 2.17. Question 3.32 (Friedman,Khomskii,Kulikov;[24]) Complete the diagram of implications (Figure 4) for regularity properties related to forcing notions at the 1 1 level.…”

“…Since the club filter is a counterexample to all the properties considered above, a natural question is the following: Question 3.33 (Friedman,Khomskii,Kulikov;[24]) Are there adequate generalisations of classical regularity properties related to forcing notions for which the club filter is regular?…”

Cichoń’s diagram for uncountable cardinals

“…We finally note that instead of generalized Sacks forcing we may consider generalized Silver forcing (see e.g. [FKK,Example 3.2] for this forcing notion) and add many κ-Silver functions either with a κ-support product or a κ-support iteration. It is not difficult to see that this has the same effect on our cardinals, that is, d κ (∈ * ) is increased to 2 κ and d h (∈ * ) stays at κ + where h is the power set function.…”

Long reals

“…In particular, we would like to draw attention both on the sometimes surprising results emerging in this area, and on the remarkably wide variety of open problems and lines for further research that are consequently revealed. Recent work, by various people, on generalized Baire spaces and related set-theoretic issues (see, for instance, Friedman, Khomskii and Kulikov [15] and Khomskii, Laguzzi, Löwe and Sharankou [23]) should be thought as complementary in this direction, possibly with important underlying connections waiting to surface.…”