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On the complemented subspaces of the Schreier spaces

I. Gasparis, D. Leung (2000)

Studia Mathematica

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space X ξ generated by subsequences ( e l n ξ ) and ( e m n ξ ) , respectively, of the natural Schauder basis ( e n ξ ) of X ξ are isomorphic if and only if ( e l n ξ ) and ( e m n ξ ) are equivalent. Further, X ξ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of ( e n ξ ) . It is also shown that there exists a complemented subspace spanned by a block basis of ( e n ξ ) , which is not isomorphic to a subspace generated by a subsequence of ( e n ζ ) , for every 0 ζ ξ ....

On the complexity of subspaces of S ω

Carlos Uzcátegui (2003)

Fundamenta Mathematicae

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set 2 X (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space S ω is F σ δ . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of S ω . We show that S ω has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...

On the complexity of sums of Dirichlet measures

Sylvain Kahane (1993)

Annales de l'institut Fourier

Let M be the set of all Dirichlet measures on the unit circle. We prove that M + M is a non Borel analytic set for the weak* topology and that M + M is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates M + M from D (or even L 0 ) , the set of all measures singular with respect to every measure in M . This extends results of Kaufman, Kechris and Lyons about D and H and gives many examples of non Borel analytic sets.

On the construction of t-norms (t-conorms) by using interior (closure) operator on bounded lattices

Emel Aşıcı (2022)

Kybernetika

Recently, the topic of construction methods for triangular norms (triangular conorms), uninorms, nullnorms, etc. has been studied widely. In this paper, we propose construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices by using interior and closure operators, respectively. Thus, we obtain some proposed methods given by Ertuğrul, Karaçal, Mesiar [15] and Çaylı [8] as results. Also, we give some illustrative examples. Finally, we conclude that the...

On the constructions of t-norms and t-conorms on some special classes of bounded lattices

Emel Aşıcı (2021)

Kybernetika

Recently, the topic related to the construction of triangular norms and triangular conorms on bounded lattices using ordinal sums has been extensively studied. In this paper, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an appropriate bounded lattice. Also, we give some illustrative examples for clarity. Then, we show that a new construction method can be generalized by induction to a modified ordinal sum for triangular norms and triangular conorms on...

On the Converse of Caristi's Fixed Point Theorem

Szymon Głąb (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Let X be a nonempty set of cardinality at most 2 and T be a selfmap of X. Our main theorem says that if each periodic point of T is a fixed point under T, and T has a fixed point, then there exist a metric d on X and a lower semicontinuous map ϕ :X→ ℝ ₊ such that d(x,Tx) ≤ ϕ(x) - ϕ(Tx) for all x∈ X, and (X,d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.

On the diameter of the Banach-Mazur set

Gilles Godefroy (2010)

Czechoslovak Mathematical Journal

On every subspace of l ( ) which contains an uncountable ω -independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of l ( ) is infinite. This provides a partial answer to a question asked by Johnson and Odell.

On the difference property of Borel measurable functions

Hiroshi Fujita, Tamás Mátrai (2010)

Fundamenta Mathematicae

If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.

On the direct product of uninorms on bounded lattices

Emel Aşıcı, Radko Mesiar (2021)

Kybernetika

In this paper, we study on the direct product of uninorms on bounded lattices. Also, we define an order induced by uninorms which are a direct product of two uninorms on bounded lattices and properties of introduced order are deeply investigated. Moreover, we obtain some results concerning orders induced by uninorms acting on the unit interval [ 0 , 1 ] .

On the existence of a σ -closed dense subset

Jindřich Zapletal (2010)

Commentationes Mathematicae Universitatis Carolinae

It is consistent with the axioms of set theory that there are two co-dense partial orders, one of them σ -closed and the other one without a σ -closed dense subset.

On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF

Eleftherios Tachtsis (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

In ZF (i.e. Zermelo-Fraenkel set theory without the Axiom of Choice AC), we investigate the relationship between UF(ω) (there exists a free ultrafilter on ω) and the statements "there exists a free ultrafilter on every Russell-set" and "there exists a Russell-set A and a free ultrafilter ℱ on A". We establish the following results: 1. UF(ω) implies that there exists a free ultrafilter on every Russell-set. The implication is not reversible in ZF. 2. The statement...

On the existence of group localizations under large-cardinal axioms.

Carles Casacuberta, Dirk Scevenels (2001)

RACSAM

Uno de los problemas abiertos más antiguos de la teoría de grupos categórica es si todo par ortogonal (formado por una clase de grupos y una clase de homomorfismos que se determinan mutuamente por ortogonalidad en el sentido de Freyd-Kelly), se halla asociado a un funtor de localización. Se sabe que esto es cierto si se acepta la validez de un cierto axioma de cardinales grandes (el principio de Vopenka), pero no se conoce ninguna demostración mediante los axiomas ordinarios (ZFC) de la teoría de...

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