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A C(K) Banach space which does not have the Schroeder-Bernstein property

Piotr Koszmider (2012)

Studia Mathematica

We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...

Boolean Rings that are Baire Spaces

Haydon, R. (2001)

Serdica Mathematical Journal

∗ The present article was originally submitted for the second volume of Murcia Seminar on Functional Analysis (1989). Unfortunately it has been not possible to continue with Murcia Seminar publication anymore. For historical reasons the present vesion correspond with the original one.Weak completeness properties of Boolean rings are related to the property of being a Baire space (when suitably topologised) and to renorming properties of the Banach spaces of continuous functions on the corresponding...


B. Banaschewski, A. Pultr (1996)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Booleanization of uniform frames

Bernhard Banaschewski, Aleš Pultr (1996)

Commentationes Mathematicae Universitatis Carolinae

Booleanization of frames or uniform frames, which is not functorial under the basic choice of morphisms, becomes functorial in the categories with weakly open homomorphisms or weakly open uniform homomorphisms. Then, the construction becomes a reflection. In the uniform case, moreover, it also has a left adjoint. In connection with this, certain dual equivalences concerning uniform spaces and uniform frames arise.

Chains and antichains in Boolean algebras

M. Losada, Stevo Todorčević (2000)

Fundamenta Mathematicae

We give an affirmative answer to problem DJ from Fremlin’s list [8] which asks whether M A ω 1 implies that every uncountable Boolean algebra has an uncountable set of pairwise incomparable elements.

Combinatorics of dense subsets of the rationals

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

Fundamenta Mathematicae

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.

Constructions of thin-tall Boolean spaces.

Juan Carlos Martínez (2003)

Revista Matemática Complutense

This is an expository paper about constructions of locally compact, Hausdorff, scattered spaces whose Cantor-Bendixson height has cardinality greater than their Cantor-Bendixson width.

Contribución al estudio de los anillos reticulados y f-anillos.

Joan Trías Pairó (1979)


This paper deals with ordered rings and f-rings. Some relations between classes of ideals are obtained. The idea of subunity allows us to study the possibility of embedding the ring in a unitary f-ring. The Boolean algebras of idempotents and lattice-isometries in an f-ring are studied. We give geometric characterizations of the l-isometries and obtain, in the projectable case, that the Stone space of the Boolean algebra of l-isometries is homeomorphic to the space of minimal prime ideals with the...

Dominating analytic families

Anastasis Kamburelis (1998)

Fundamenta Mathematicae

Let A be an analytic family of sequences of sets of integers. We show that either A is dominated or it contains a continuum of almost disjoint sequences. From this we obtain a theorem by Shelah that a Suslin c.c.c. forcing adds a Cohen real if it adds an unbounded real.

Egoroff, σ, and convergence properties in some archimedean vector lattices

A. W. Hager, J. van Mill (2015)

Studia Mathematica

An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each a n c o n A there are λ n ( 0 , ) and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...

Equimorphy in varieties of distributive double p -algebras

Václav Koubek, Jiří Sichler (1998)

Czechoslovak Mathematical Journal

Any finitely generated regular variety 𝕍 of distributive double p -algebras is finitely determined, meaning that for some finite cardinal n ( 𝕍 ) , any subclass S 𝕍 of algebras with isomorphic endomorphism monoids has fewer than n ( 𝕍 ) pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double p -algebras...

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