The Experts below are selected from a list of 234 Experts worldwide ranked by ideXlab platform
Juris Steprāns - One of the best experts on this subject based on the ideXlab platform.
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Splitting families and complete separability
Canadian Mathematical Bulletin, 2014Co-Authors: Heike Mildenberger, Dilip Raghavan, Juris SteprānsAbstract:We answer a question from Raghavan and Stepr{\=a}ns' paper on weakly tight families by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost Disjoint Family under $\s \leq \a$, partially answering a question of Shelah.
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On Weakly Tight Families
Canadian Journal of Mathematics, 2012Co-Authors: Dilip Raghavan, Juris SteprānsAbstract:Using ideas from Shelah’s recent proof that a completely separable maximal almost Disjoint Family exists when c < אω , we construct a weakly tight Family under the hypothesis s ≤ b < אω . The case when s < b is handled in ZFC and does not require b < אω , while an additional PCF type hypothesis, which holds when b < אω , is used to treat the case s = b. The notion of a weakly tight Family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost Disjoint Family. It was introduced by Hrusak and Garcia Ferreira [8], who applied it to the Katetov order on almost Disjoint families.
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On weakly tight families
arXiv: Logic, 2010Co-Authors: Dilip Raghavan, Juris SteprānsAbstract:Using ideas from Shelah's recent proof that a completely separable maximal almost Disjoint Family exists when $\c < {\aleph}_{\omega}$, we construct a weakly tight Family under the hypothesis $\s \leq \b < {\aleph}_{\omega}$. The case when $\s < \b$ is handled in $\ZFC$ and does not require $\b < {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\b < {\aleph}_{\omega}$ is used to treat the case $\s = \b$. The notion of a weakly tight Family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost Disjoint Family. It was introduced by Hru{\v{s}}{\'a}k and Garc{\'{\i}}a Ferreira \cite{Hr1}, who applied it to the Kat\'etov order on almost Disjoint families.
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Homogeneous almost Disjoint families
algebra universalis, 1994Co-Authors: Sharon Shelah, Juris SteprānsAbstract:An almost Disjoint Family is constructed which is isomorphic to any almost Disjoint Family which can be constructed from it by taking subsets and finite unions. This is applied to the construction of a Boolean algebra with related properties.
Dilip Raghavan - One of the best experts on this subject based on the ideXlab platform.
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Splitting families and complete separability
Canadian Mathematical Bulletin, 2014Co-Authors: Heike Mildenberger, Dilip Raghavan, Juris SteprānsAbstract:We answer a question from Raghavan and Stepr{\=a}ns' paper on weakly tight families by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost Disjoint Family under $\s \leq \a$, partially answering a question of Shelah.
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On Weakly Tight Families
Canadian Journal of Mathematics, 2012Co-Authors: Dilip Raghavan, Juris SteprānsAbstract:Using ideas from Shelah’s recent proof that a completely separable maximal almost Disjoint Family exists when c < אω , we construct a weakly tight Family under the hypothesis s ≤ b < אω . The case when s < b is handled in ZFC and does not require b < אω , while an additional PCF type hypothesis, which holds when b < אω , is used to treat the case s = b. The notion of a weakly tight Family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost Disjoint Family. It was introduced by Hrusak and Garcia Ferreira [8], who applied it to the Katetov order on almost Disjoint families.
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A model with no strongly separable almost Disjoint families
Israel Journal of Mathematics, 2012Co-Authors: Dilip RaghavanAbstract:We answer a question of Shelah and Steprāns [6] by producing a model of ZFC where there are no strongly separable almost Disjoint families. The notion of a strongly separable almost Disjoint Family is a natural variation on the well known notion of a completely separable almost Disjoint Family, and is closely related to the metrization problem for countable Fréchet groups.
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Comparing the closed almost Disjointness and dominating numbers
Fundamenta Mathematicae, 2012Co-Authors: Dilip Raghavan, Saharon ShelahAbstract:We prove that if there is a dominating Family of size א1, then there are א1 many compact subsets of ω whose union is a maximal almost Disjoint Family of functions that is also maximal with respect to infinite partial functions.
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Comparing the closed almost Disjointness and dominating numbers
arXiv: Logic, 2011Co-Authors: Dilip Raghavan, Saharon ShelahAbstract:We prove that if there is a dominating Family of size ${\aleph}_{1}$, then there is are ${\aleph}_{1}$ many compact subsets of ${\omega}^{\omega}$ whose union is a maximal almost Disjoint Family of functions that is also maximal with respect to infinite partial functions.
Niels Jakob Laustsen - One of the best experts on this subject based on the ideXlab platform.
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A Banach space induced by an almost Disjoint Family, admitting only few operators and decompositions
Advances in Mathematics, 2021Co-Authors: Piotr Koszmider, Niels Jakob LaustsenAbstract:Abstract We consider the closed subspace of l ∞ generated by c 0 and the characteristic functions of elements of an uncountable, almost Disjoint Family A of infinite subsets of N . This Banach space has the form C 0 ( K A ) for a locally compact Hausdorff space K A that is known under many names, including Ψ-space and Isbell–Mrowka space. We construct an uncountable, almost Disjoint Family A such that the algebra of all bounded linear operators on C 0 ( K A ) is as small as possible in the precise sense that every bounded linear operator on C 0 ( K A ) is the sum of a scalar multiple of the identity and an operator that factors through c 0 (which in this case is equivalent to having separable range). This implies that C 0 ( K A ) has the fewest possible decompositions: whenever C 0 ( K A ) is written as the direct sum of two infinite-dimensional Banach spaces X and Y , either X is isomorphic to C 0 ( K A ) and Y to c 0 , or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space C 0 ( K A ) concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms. To exploit the perfect set property for Borel sets as in the classical construction of an almost Disjoint Family by Mrowka, we need to deal with N × N matrices rather than with the usual partitioners of an almost Disjoint Family. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to van Engelen, Kunen and Miller concerning Borel subsets of the square.
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A Banach space induced by an almost Disjoint Family, admitting only few operators and decompositions
arXiv: Functional Analysis, 2020Co-Authors: Piotr Koszmider, Niels Jakob LaustsenAbstract:We consider the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost Disjoint Family $\mathcal A$ of infinite subsets of $\mathbb N$. This Banach space has the form $C_0(K_{\mathcal A})$ for a locally compact Hausdorff space $K_{\mathcal A}$ that is known under many names, such as $\Psi$-space and Isbell--Mrowka space. We construct an uncountable, almost Disjoint Family ${\mathcal A}$ such that the Banach algebra of all bounded linear operators on $C_0(K_{\mathcal A})$ is as small as possible in the sense that every bounded linear operator on $C_0(K_{\mathcal A})$ is the sum of a scalar multiple of the identity and an operator that factors through $c_0$ (which in this case is equivalent to having separable range). This implies that $C_0(K_{\mathcal A})$ has the fewest possible decompositions: whenever $C_0(K_{\mathcal A})=X\oplus Y$ with $dim({X})=\infty$, $dim({Y})=\infty$, either ${X}$ is isomorphic to $C_0(K_{\mathcal A})$ and ${Y}$ to $c_0$, or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space $C_0(K_{\mathcal A})$ concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms. To exploit the perfect set property for Borel sets as in the classical construction of an almost Disjoint Family of Mrowka we need to deal with $\mathbb N \times \mathbb N$-matrices rather than with the usual partitioners. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to F. van Engelen, K. Kunen and A. Miller concerning Borel subsets of the square.
Paul J. Szeptycki - One of the best experts on this subject based on the ideXlab platform.
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transversals for strongly almost Disjoint families
Proceedings of the American Mathematical Society, 2007Co-Authors: Paul J. SzeptyckiAbstract:For a Family of sets A, and a set X, X is said to be a transversal of A if X C ∪ A and |a ∩ X | = 1 for each a ∈ A. X is said to be a Bernstein set for A if ∅ ≠ a ∩ X ≠ a for each a ∈ A. Erdos and Hajnal first studied when an almost Disjoint Family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a Family of sets A is said to admit a σ-transversal if A can be written as A = ∪{A n : n ∈ ω} such that each An admits a transversal. We study the question of when an almost Disjoint Family admits a σ-transversal and related questions.
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Spaces of continuous functions defined on Mrówka spaces
Topology and its Applications, 2005Co-Authors: Michael Hrušák, Paul J. Szeptycki, Angel Tamariz-mascarúaAbstract:Abstract We prove that for a maximal almost Disjoint Family A on ω, the space C p ( Ψ ( A ) , 2 ω ) of continuous Cantor-valued functions with the pointwise convergence topology defined on the Mrowka space Ψ ( A ) is not normal. Using CH we construct a maximal almost Disjoint Family A for which the space C p ( Ψ ( A ) , 2 ) of continuous { 0 , 1 } -valued functions defined on Ψ ( A ) is Lindelof. These theorems improve some results due to Dow and Simon in [Spaces of continuous functions over a Ψ-space, Preprint]. We also prove that this space C p ( Ψ ( A ) , 2 ) = X is a Michael space; that is, X n is Lindelof for every n ∈ N and neither X ω nor X × ω ω are normal. Moreover, we prove that for every uncountable almost Disjoint Family A on ω and every compactification b Ψ ( A ) of Ψ ( A ) , the space C p ( b Ψ ( A ) , 2 ω ) is not normal.
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Soft almost Disjoint families
Proceedings of the American Mathematical Society, 2002Co-Authors: Paul J. SzeptyckiAbstract:An almost Disjoint Family A is said to be soft if there is an infinite set that meets each a ∈ A in a nonempty but finite set. We consider the associated cardinal invariant defined to be the minimal cardinality of an almost Disjoint Family that is not soft. We show that this cardinal coincides with J. Brendle's cardinal ap.
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Countable metacompactness in Ψ-spaces
Proceedings of the American Mathematical Society, 1994Co-Authors: Paul J. SzeptyckiAbstract:We prove under a variety of assumptions including c = R2 that, for every maximal almost Disjoint Family A? of countable subsets of coI, T(.V) is not countably metacompact. In addition, a first countable, countably metacompact, regular space with a closed discrete set which is not a G35 is constructed from the mutually consistent assumptions that b = c01 and there can exist a Q-set.
Michael Hrušák - One of the best experts on this subject based on the ideXlab platform.
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Maximal almost Disjoint families and pseudocompactness of hyperspaces
arXiv: General Topology, 2020Co-Authors: Osvaldo Guzmán, Michael Hrušák, Vinicius De Oliveira Rodrigues, Stevo Todorcevic, Artur Hideyuki TomitaAbstract:We show that all maximal almost Disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(\omega)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost Disjoint Family whose hyperspace is pseudocompact and prove that consistently such families do not exist \emph{genericaly}, by constructing a consistent example of a maximal almost Disjoint Family $\mathcal A$ of size less than $\mathfrak c$ whose hyperspace is not pseudocompact.
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Fréchet-like properties and almost Disjoint families
Topology and its Applications, 2020Co-Authors: César Corral, Michael HrušákAbstract:Abstract We study the relationship between α i properties and strong Frechet-like properties in Ψ-spaces associated to almost Disjoint families. In particular, we prove that under some mild assumptions (e.g. c ≤ ℵ 2 ) there is an almost Disjoint Family A such that Ψ ( A ) is Frechet, α 3 and not bisequential, answering a question of G. Gruenhage.
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On $\mathbb R$-embeddability of almost Disjoint families and Akemann-Doner C*-algebras
arXiv: Logic, 2019Co-Authors: Osvaldo Guzmán, Michael Hrušák, Piotr KoszmiderAbstract:An almost Disjoint Family $\mathcal A$ of subsets of $\mathbb N$ is said to be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow \mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to distinct reals for distinct $A\in \mathcal A$. It is well known that almost Disjoint families which have few separations, such as Luzin families, are not $\mathbb R$-embeddable. We study extraction principles related to $\mathbb R$-embeddability and separation properties of almost Disjoint families of $\mathbb N$ as well as their limitations. An extraction principle whose consistency is our main result is: every almost Disjoint Family of size continuum contains an $\mathbb R$-embeddable subFamily of size continuum. It is true in the Sacks model. The Cohen model serves to show that the above principle does not follow from the fact that every almost Disjoint Family of size continuum has two separated subfamilies of size continuum. We also construct in ZFC an almost Disjoint Family, where no two uncountable subfamilies can be separated but always a countable subFamily can be separated from any Disjoint subFamily. Using a refinement of the $\mathbb R$-embeddability property called a controlled $\mathbb R$-embedding property we obtain the following results concerning Akemann-Doner C*-algebras which are induced by uncountable almost Disjoint families: a) In ZFC there are Akemann-Doner C*-algebras of density $\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$, b) It is independent from ZFC whether there is an Akemann-Doner algebra of density $\mathfrak c$ with no nonseparable commutative subalgebra. This completes an earlier result that there is in ZFC an Akemann-Doner algebra of density $\omega_1$ with no nonseparable commutative subalgebra.
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Topology of Mrówka-Isbell Spaces
Pseudocompact Topological Spaces, 2018Co-Authors: Fernando Hernández-hernández, Michael HrušákAbstract:An infinite Family \(\mathscr {A}\) of infinite subsets of the natural numbers, \(\omega \), is almost Disjoint (AD) if the intersection of any two distinct elements of \(\mathscr {A}\) is finite. It is maximal almost Disjoint (MAD) if given an infinite \(X\subset \omega \) there is an \(A\in \mathscr {A}\) such that \(|A\cap X|=\omega \), in other words, if the Family \(\mathscr {A}\) is not included in any larger almost Disjoint Family.
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Spaces of continuous functions defined on Mrówka spaces
Topology and its Applications, 2005Co-Authors: Michael Hrušák, Paul J. Szeptycki, Angel Tamariz-mascarúaAbstract:Abstract We prove that for a maximal almost Disjoint Family A on ω, the space C p ( Ψ ( A ) , 2 ω ) of continuous Cantor-valued functions with the pointwise convergence topology defined on the Mrowka space Ψ ( A ) is not normal. Using CH we construct a maximal almost Disjoint Family A for which the space C p ( Ψ ( A ) , 2 ) of continuous { 0 , 1 } -valued functions defined on Ψ ( A ) is Lindelof. These theorems improve some results due to Dow and Simon in [Spaces of continuous functions over a Ψ-space, Preprint]. We also prove that this space C p ( Ψ ( A ) , 2 ) = X is a Michael space; that is, X n is Lindelof for every n ∈ N and neither X ω nor X × ω ω are normal. Moreover, we prove that for every uncountable almost Disjoint Family A on ω and every compactification b Ψ ( A ) of Ψ ( A ) , the space C p ( b Ψ ( A ) , 2 ω ) is not normal.