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An Isomorphic Classification of C ( 2 × [ 0 , α ] ) Spaces

Elói Medina Galego (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces C ( 2 × [ 0 , α ] ) of all real continuous functions defined on the compact spaces 2 × [ 0 , α ] , the topological product of the Cantor cubes 2 with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently, it is relatively...

Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

Ramez Sami (1999)

Fundamenta Mathematicae

We prove the following theorem: Given a⊆ω and 1 ≤ α < ω 1 C K , if for some η < ℵ 1 and all u ∈ WO of length η, a is Σ α 0 ( u ) , then a is Σ α 0 . We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ 1 1 -Turing-determinacy implies the existence of 0 .

Applications of some strong set-theoretic axioms to locally compact T₅ and hereditarily scwH spaces

Peter J. Nyikos (2003)

Fundamenta Mathematicae

Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T₅ spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theorem (Theorem 1) that applies to all such ω₁-compact spaces and another (Theorem 4) to all such spaces of Lindelöf number ≤ ℵ₁. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact...

Borel extensions of Baire measures in ZFC

Menachem Kojman, Henryk Michalewski (2011)

Fundamenta Mathematicae

We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property ( a )

Samuel Gomes da Silva (2011)

Commentationes Mathematicae Universitatis Carolinae

In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property ( a ) . It follows that it is consistent that closed discrete subsets of a separable space X which are also relatively normal (relatively countably paracompact, relatively ( a ) ) in X are necessarily countable. There are, however, consistent examples of...

Colimit-dense subcategories

Jiří Adámek, Andrew D. Brooke-Taylor, Tim Campion, Leonid Positselski, Jiří Rosický (2019)

Commentationes Mathematicae Universitatis Carolinae

Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vopěnka’s Principle, we prove that a cocomplete category is locally presentable if and only if it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a 3 -element set is colimit-dense in 𝐒𝐞𝐭 op , and spaces of countable dimension are colimit-dense in 𝐕𝐞𝐜 op .

Condensation and large cardinals

Sy-David Friedman, Peter Holy (2011)

Fundamenta Mathematicae

We introduce two generalized condensation principles: Local Club Condensation and Stationary Condensation. We show that while Strong Condensation (a generalized condensation principle introduced by Hugh Woodin) is inconsistent with an ω₁-Erdős cardinal, Stationary Condensation and Local Club Condensation (which should be thought of as weakenings of Strong Condensation) are both consistent with ω-superstrong cardinals.

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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 κ κ 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 κ κ for uncountable regular κ is however consistent (with GCH), assuming...

Continuous tree-like scales

James Cummings (2010)

Open Mathematics

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵω is consistent with Martin’s Maximum.

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