Abstract
In this work we study cardinal invariants of the ideal of strongly porous sets on $\tothe{2}{\omega}$. We prove that $\add{SP}{} = \omega_1$, $\cof{SP}{} = \mathfrak{c}$ and that it is consistent that $\non{SP}{} < \add{\mathcal{N}}{}$, answering questions of Hru\v{s}\'ak and Zindulka. We also find a connection between strongly porous sets on $\tothe{2}{\omega}$ and the Martin number for $\sigma$-linked partial orders, and we use this connection to construct a model where all the Martin numbers for $\sigma$-$k$-linked forcings are mutually different.
Highlights
In 1967, Dolzenko [5] began the study of σ -porous sets1 and since many applications have been found
We shall study the notion of strong porosity: given a metric space X, d, a subset A ⊆ X is strongly porous if there is a p > 0 such that for any x ∈ X and any 0 < r < 1, there is y ∈ X such that Bp·r(y) ⊆ Br(x) \ A
Cardinal invariants of these σ -ideals have been studied in Brendle [3], Hrusak and Zindulka [6] and Repicky [13, 14, 15]
Summary
In 1967, Dolzenko [5] began the study of σ -porous sets and since many applications have been found. Further research about different types of porosity can be found in Rojas-Rebolledo [16], Zajıcek [17], Zapletal [18], Zeleny [20] and Zelenyand Zajıcek [21] Cardinal invariants of these σ -ideals have been studied in Brendle [3], Hrusak and Zindulka [6] and Repicky [13, 14, 15]. They proved that non(SP) < mσ-centered is consistent, that cov(SP) > cof(N ) is consistent, and that cov(SP) < c is consistent, where mσ-centered is the smallest cardinal where the Martin’s axiom for σ -centered forcings fails and N is the ideal of sets of Lebesgue measure zero. Our notation follows Bartoszynski and Judah [1]
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