### A cardinal preserving immune partition of the ordinals

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Under the assumption of the existence of sharps for reals all simply definable posets on ${\omega}_{1}$ are classified up to forcing equivalence.

We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.

We define combinatorial structures which we refer to as flat morasses, and use them to construct a Lindelöf space with points ${G}_{\delta}$ of cardinality ${\aleph}_{\omega}$, consistent with GCH. The construction reveals, it is hoped, that flat morasses are a tool worth adding to the kit of any user of set theory.

Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement "$\forall \alpha \in {\omega}^{\omega},\omega {\u2081}^{L\left(\alpha \right)}<\omega \u2081$"; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets of a Borel...

Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space ${\kappa}^{\kappa}$ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in ${\kappa}^{\kappa}$ for uncountable regular κ is however consistent (with GCH), assuming...

The discrete algebras $A$ over a commutative ring $R$ which can be realized as the full endomorphism algebra of a torsion-free $R$-module have been investigated by Dugas and Göbel under the additional set-theoretic axiom of constructibility, $V=L$. Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are rederived in a more natural topological setting and substantial generalizations to topological...

We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a ${G}_{\delta}$. In addition some nonperfect spaces with σ-disjoint bases are constructed.