It is proved, under Martin's Axiom, that all gaps in are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin's Axiom the Baire categoricity of all Δ31 non-Δ31-complete sets of reals requires a weakly compact cardinal.

We investigate cardinal invariants related to the structure of dense sets of rationals modulo the nowhere dense sets. We prove that , thus dualizing the already known [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 59–80, Theorem 3.6]. We also show the consistency of each of and . Our results answer four questions of Balcar, Hernández and Hrušák [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. (...) 183 59–80, Questions 3.11]. (shrink)

We investigate the effect of adding a single real on cardinal invariants associated with the continuum. We show:1. adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1;2. Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iteratively always force CH;3. Miller's rational perfect set forcing preserves the axiom MA.

Brendle, J., H. Judah and S. Shelah, Combinatorial properties of Hechler forcing, Annals of Pure and Applied Logic 59 185–199. Using a notion of rank for Hechler forcing we show: assuming ωV1 = ωL1, there is no real in V[d] which is eventually different from the reals in L[ d], where d is Hechler over V; adding one Hechler real makes the invariants on the left-hand side of Cichoń's diagram equal ω1 and those on the right-hand side equal 2ω and (...) produces a maximal almost disjoint family of subsets of ω of size ω1; there is no perfect set of random reals over V in V[ r][ d], where r is random over V and d Hechler over V[r], thus answering a question of the first and second authors. (shrink)

We present several naturally defined σ-ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity. We investigate set-theoretic properties of such σ-ideals.

We extend results of Elekes and Máthé on monotone Borel hulls to an abstract setting of measurable space with negligibles. This scheme yields the respective theorems in the case of category and in the cases associated with the Mendez σ-ideals on the plane. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.