Displaying similar documents to “Cardinal sequences of length < ω₂ under GCH”

Coherent ultrafilters and nonhomogeneity

Jan Starý (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We introduce the notion of a coherent P -ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a P -point on ω , and show that these ultrafilters exist generically under 𝔠 = 𝔡 . This improves the known existence result of Ketonen [On the existence of P -points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1,...

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We show that given infinite sets X , Y and a function f : X Y which is onto and n -to-one for some n , the preimage of any ultrafilter of Y under f extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms A and a finite-to-one onto function f : A ω such that for each free ultrafilter of ω its preimage under f does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact...

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let a space X be Tychonoff product α < τ X α of τ -many Tychonoff nonsingle point spaces X α . Let Suslin number of X be strictly less than the cofinality of τ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification β X . In particular, this is true if X is either R τ or ω τ and a cardinal τ is infinite and not countably cofinal.

On the solvability of systems of linear equations over the ring of integers

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We investigate the question whether a system ( E i ) i I of homogeneous linear equations over is non-trivially solvable in provided that each subsystem ( E j ) j J with | J | c is non-trivially solvable in where c is a fixed cardinal number such that c < | I | . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., I being finite). (b) The answer is ‘No’ in the denumerable case (i.e., | I | = 0 and c a natural number). (c) The answer in case that I is uncountable and c 0 is ‘No...

Spaces with property ( D C ( ω 1 ) )

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that if X is a first countable space with property ( D C ( ω 1 ) ) and with a G δ -diagonal then the cardinality of X is at most 𝔠 . We also show that if X is a first countable, DCCC, normal space then the extent of X is at most 𝔠 .

Cardinal invariants for κ-box products: weight, density character and Suslin number

W. W. Comfort, Ivan S. Gotchev

Similarity:

The symbol ( X I ) κ (with κ ≥ ω) denotes the space X I : = i I X i with the κ-box topology; this has as base all sets of the form U = i I U i with U i open in X i and with | i I : U i X i | < κ . The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each X i contains the discrete space 0,1 and satisfies w ( X i ) α , then w ( X κ ) = α < κ . Theorem 4.3.2. If ω κ | I | 2 α and X = ( D ( α ) ) I with D(α) discrete, |D(α)| = α, then d ( ( X I ) κ ) = α < κ . Corollaries 5.2.32(a)...

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

Similarity:

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class 𝒮 of morphisms in a locally presentable category 𝒞 of structures, the orthogonal class of objects...

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if σ D M is a support, M being an inner model of ZFC, and 𝒫 ( D σ ) M = r ` ` σ with r M , then r determines a preorder " " of D such that σ becomes a filter on ( D , ) generic over M . We show that if the relation r is replaced by a function 𝒫 ( D σ ) M = f - 1 ( σ ) , then there exists an equivalence relation " " on D and a partial order on D / such that D / is a complete Boolean algebra, σ / is a generic filter and [ f ( u ) ] = - ( u / ) for...

Coloring Cantor sets and resolvability of pseudocompact spaces

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let us denote by Φ ( λ , μ ) the statement that 𝔹 ( λ ) = D ( λ ) ω , i.e. the Baire space of weight λ , has a coloring with μ colors such that every homeomorphic copy of the Cantor set in 𝔹 ( λ ) picks up all the μ colors. We call a space X π -regular if it is Hausdorff and for every nonempty open set U in X there is a nonempty open set V such that V ¯ U . We recall that a space X is called feebly compact if every locally finite collection of open sets in X is finite. A Tychonov space is pseudocompact if and...

On certain non-constructive properties of infinite-dimensional vector spaces

Eleftherios Tachtsis (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In set theory without the axiom of choice ( AC ), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC LO (AC for linearly ordered families of nonempty sets)—and hence AC WO (AC for well-ordered families of nonempty sets)— DC ( < κ ) (where κ is an uncountable regular cardinal), and “for every infinite set X , there is a bijection f : X { 0 , 1 } × X ”, implies the statement “there exists a field F such that...

Exponential domination in function spaces

Vladimir Vladimirovich Tkachuk (2020)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Given a Tychonoff space X and an infinite cardinal κ , we prove that exponential κ -domination in X is equivalent to exponential κ -cofinality of C p ( X ) . On the other hand, exponential κ -cofinality of X is equivalent to exponential κ -domination in C p ( X ) . We show that every exponentially κ -cofinal space X has a κ + -small diagonal; besides, if X is κ -stable, then n w ( X ) κ . In particular, any compact exponentially κ -cofinal space has weight not exceeding κ . We also establish that any exponentially κ -cofinal...

On ordinals accessible by infinitary languages

Saharon Shelah, Pauli Väisänen, Jouko Väänänen (2005)

Fundamenta Mathematicae

Similarity:

Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of L λ ω , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with D , a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where D ' , ' is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L λ ω is exactly δ ( λ ) . It was proved in [BK71] that if ℵ₀ < λ < κ a r e r e g u l a r c a r d i n a l n u m b e r s , t h e n t h e r e i s a f o r c i n g e x t e n s i o n , p r e s e r v i n g c o f i n a l i t i e s , s u c h t h a t i n t h e e x t e n s i o n 2λ = κ a n d δ ( λ ) < λ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e < λ < θ κ a r e c a r d i n a l n u m b e r s s u c h t h a t λ < λ = λ ; ∙ cf(θ) ≥ λ⁺ and μ λ < θ whenever μ < θ; ∙ κ λ = κ . Then there...

Selectors of discrete coarse spaces

Igor Protasov (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Given a coarse space ( X , ) with the bornology of bounded subsets, we extend the coarse structure from X × X to the natural coarse structure on ( { } ) × ( { } ) and say that a macro-uniform mapping f : ( { } ) X (or f : [ X ] 2 X ) is a selector (or 2-selector) of ( X , ) if f ( A ) A for each A { } ( A [ X ] 2 , respectively). We prove that a discrete coarse space ( X , ) admits a selector if and only if ( X , ) admits a 2-selector if and only if there exists a linear order “ " on X such that the family of intervals { [ a , b ] : a , b X , a b } is a base for the bornology .

Combinatorics of dense subsets of the rationals

B. Balcar, F. Hernández-Hernández, M. Hrušák (2004)

Fundamenta Mathematicae

Similarity:

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants , , , , , describing properties of Dense(ℚ). These invariants satisfy ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ . W e c o m p a r e t h e m w i t h t h e i r a n a l o g u e s i n t h e w e l l s t u d i e d B o o l e a n a l g e b r a ( ω ) / f i n . W e s h o w t h a t ℚ = p , ℚ = t a n d ℚ = i , w h e r e a s ℚ > h a n d ℚ > r a r e b o t h s h o w n t o b e r e l a t i v e l y c o n s i s t e n t w i t h Z F C . W e a l s o i n v e s t i g a t e c o m b i n a t o r i c s o f t h e i d e a l n w d o f n o w h e r e d e n s e s u b s e t s o f , . I n p a r t i c u l a r , w e s h o w t h a t non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

( 0 , 1 ) -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

Similarity:

Let m and n be positive integers, and let R = ( r 1 , ... , r m ) and S = ( s 1 , ... , s n ) be nonnegative integral vectors. Let A ( R , S ) be the set of all m × n ( 0 , 1 ) -matrices with row sum vector R and column vector S . Let R and S be nonincreasing, and let F ( R ) be the m × n ( 0 , 1 ) -matrix, where for each i , the i th row of F ( R , S ) consists of r i 1’s followed by ( n - r i ) 0’s. Let A A ( R , S ) . The discrepancy of A, disc ( A ) , is the number of positions in which F ( R ) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over...