Games on base matrices
Using a game characterization of distributivity, we show that base matrices for $\mathcal{P}(ω)/\text{fin}$ of regular height larger than $\mathfrak{h}$ necessarily have maximal branches which are not cofinal.
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arXiv subjects
Marlene Koelbing
Publications and source records attributed to Marlene Koelbing.
On heights of distributivity matrices
We construct a model in which there exists a distributivity matrix of regular height $λ$ larger than $\mathfrak{h}$; both $λ= \mathfrak{c}$ and $λ< \mathfrak{c}$ are possible. A distributivity matrix is a refining system of mad families without common refinement. Of particular interest in our proof is the preservation of $\mathcal{B}$-Canjarness.
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Ideal Topologies in Higher Descriptive Set Theory
We investigate generalizations of the topology of the higher Cantor space on $2^κ$, based on arbitrary ideals rather than the bounded ideal on $κ$. Our main focus is on the topology induced by the nonstationary ideal, and we call this topology the nonstationary topology, or also the Edinburgh topology on $2^κ$. It may be of independent interest that as a side result, we show $κ$-Silver forcing to satisfy a strong form of Axiom $A$ not only if $κ$ is inaccessible (which is well-known), but also under the assumption $\diamondsuit_κ$.
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Laver Trees in the Generalized Baire Space
We prove that any suitable generalization of Laver forcing to the space $ κ^κ$, for uncountable regular $κ$, necessarily adds a Cohen $κ$-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if $ κ^{<κ}=κ$, then every $<κ$-distributive tree forcing on $κ^κ$ adding a dominating $κ$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $κ$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08140
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