The Baire Category of Subsequences and Permutations which preserve Limit Points

Let $$\mathcal {I}$$ I be a meager ideal on $$\mathbf {N}$$ N . We show that if x is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of x which preserve the set of $$\mathcal {I}$$ I -cluster points of x is topologically large if and only if every ordinary limit point of x is also an $$\mathcal {I}$$ I -cluster point of x. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. 263 (2019), 221–229]. As an application, if x is a sequence with values in a first countable compact space which is $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ , then the set of subsequences [resp. permutations] which are $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ is topologically large if and only if x is convergent to $$\ell $$ ℓ in the ordinary sense. Analogous results hold for $$\mathcal {I}$$ I -limit points, provided $$\mathcal {I}$$ I is an analytic P-ideal.

No Tukey reduction of Lebesgue null to Silver null sets

We prove that consistently the Lebesgue null ideal is not Tukey reducible to the Silver null ideal. This contrasts with the situation for the meager ideal which, by a recent result of the author, Spinas [Silver trees and Cohen reals, Israel J. Math. 211 (2016) 473–480] is Tukey reducible to the Silver ideal.

Splitting families and the Noetherian type of βω / ω

Extending some results of Malykhin, we prove several independence results about base properties of βω / ω and its powers, especially the Noetherian type Nt(βω / ω), the least κ for which βω / ω has a base that is κ-like with respect to containment. For example, Nt(βω / ω) is at least s, but can consistently be ω1, c, c+, or strictly between ω1 and c. Nt(βω / ω) is also consistently less than the additivity of the meager ideal. Nt(βω / ω) is closely related to the existence of special kinds of splitting families.