Displaying similar documents to “A compact Hausdorff topology that is a T₁-complement of itself”

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when X M = X , where X M is the elementary submodel topology on X ∩ M, especially in the case when X M is compact.

Connectedness of some rings of quotients of C ( X ) with the m -topology

F. Azarpanah, M. Paimann, A. R. Salehi (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this article we define the m -topology on some rings of quotients of C ( X ) . Using this, we equip the classical ring of quotients q ( X ) of C ( X ) with the m -topology and we show that C ( X ) with the r -topology is in fact a subspace of q ( X ) with the m -topology. Characterization of the components of rings of quotients of C ( X ) is given and using this, it turns out that q ( X ) with the m -topology is connected if and only if X is a pseudocompact almost P -space, if and only if C ( X ) with r -topology is connected. We also...

A hit-and-miss topology for 2 X , Cₙ(X) and Fₙ(X)

Benjamín Espinoza, Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano (2009)

Colloquium Mathematicae

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A hit-and-miss topology ( τ H M ) is defined for the hyperspaces 2 X , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between τ H M and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets J 0 , σ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of J 0 , σ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS ϕ σ , α n , k is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

Topological properties of some spaces of continuous operators

Marian Nowak (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C b ( X , E ) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space L β ( C b ( X , E ) , F ) of all ( β , | | · | | F ) -continuous linear operators from C b ( X , E ) to F, equipped with the topology τ s of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize τ s -compact subsets of L β ( C b ( X , E ) , F ) in terms of properties of the corresponding sets of the representing...

On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets

Ludwik Jaksztas (2011)

Fundamenta Mathematicae

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Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map g σ . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set J 0 , σ is continuous at σ₀ as the function of the parameter σ ¯ if and only if H D ( J 0 , σ ) 4 / 3 . Since H D ( J 0 , σ ) > 4 / 3 on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of H D ( J 0 , σ ) on an open and dense subset of...

An irrational problem

Franklin D. Tall (2002)

Fundamenta Mathematicae

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Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define X M to be X ∩ M with topology generated by U M : U M . Suppose X M is homeomorphic to the irrationals; must X = X M ? We have partial results. We also answer a question of Gruenhage by showing that if X M is homeomorphic to the “Long Cantor Set”, then X = X M .

On the Hausdorff Dimension of Topological Subspaces

Tomasz Szarek, Maciej Ślęczka (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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It is shown that every Polish space X with d i m T X d admits a compact subspace Y such that d i m H Y d where d i m T and d i m H denote the topological and Hausdorff dimensions, respectively.

A new Lindelöf space with points G δ

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove that * implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 1 which has points G δ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

Interaction between cellularity of Alexandroff spaces and entropy of generalized shift maps

Fatemah Ayatollah Zadeh Shirazi, Sahar Karimzadeh Dolatabad, Sara Shamloo (2016)

Commentationes Mathematicae Universitatis Carolinae

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In the following text for a discrete finite nonempty set K and a self-map ϕ : X X we investigate interaction between different entropies of generalized shift σ ϕ : K X K X , ( x α ) α X ( x ϕ ( α ) ) α X and cellularities of some Alexandroff topologies on X .

Homeomorphism groups of Sierpiński carpets and Erdős space

Jan J. Dijkstra, Dave Visser (2010)

Fundamenta Mathematicae

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Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . Let M n + 1 , n ∈ ℕ, be the n-dimensional Menger continuum in n + 1 , also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of M n + 1 . We consider the topological group...

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

W. W. Comfort, Ivan S. Gotchev

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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)...

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of C k ( X ) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space C k ( X ) is Ascoli iff C k ( X ) is a k -space iff X is locally compact. Moreover, C k ( X ) endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials P c ( z ) = z d + c for c ∈ ℂ. Denote by J c the Julia set of P c and let d = c | J c i s c o n n e c t e d be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c d : those for which the critical point 0 is not recurrent by P c and without parabolic cycles. The Hausdorff dimension of J c , denoted by H D ( J c ) , does not depend continuously on c at such c d ; on the other hand the function c H D ( J c ) is analytic in - d . Our first result asserts that there is still some...

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish L C n - 1 -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

On biorthogonal systems whose functionals are finitely supported

Christina Brech, Piotr Koszmider (2011)

Fundamenta Mathematicae

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We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space K 2 n such that C ( K 2 n ) has no uncountable (semi)biorthogonal sequence ( f ξ , μ ξ ) ξ ω where μ ξ ’s are atomic measures with supports consisting of at most 2n-1 points of K 2 n , but has biorthogonal systems ( f ξ , μ ξ ) ξ ω where μ ξ ’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number α 2 , a topological group G such that G γ is countably compact for all cardinals γ < α, but G α is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under M A c o u n t a b l e . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from M A c o u n t a b l e . However, the question has...

A two-dimensional univoque set

Martijn de Vrie, Vilmos Komornik (2011)

Fundamenta Mathematicae

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Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form x = i = 1 c i q - i with integer coefficients c i satisfying 0 c i < q , i ≥ 1. In this case we say that ( c i ) = c c . . . is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also...

The AR-Property of the spaces of closed convex sets

Katsuro Sakai, Masato Yaguchi (2006)

Colloquium Mathematicae

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Let C o n v H ( X ) , C o n v A W ( X ) and C o n v W ( X ) be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of C o n v H ( X ) and the space C o n v A W ( X ) are AR. In case X is separable, C o n v W ( X ) is locally path-connected.

Univoque sets for real numbers

Fan Lü, Bo Tan, Jun Wu (2014)

Fundamenta Mathematicae

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For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with x i 0 , 1 . We prove that for any x ∈ (0,1), (x) contains a sequence β k k 1 increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure ( x ) ¯ are nowhere dense.

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

On Hattori spaces

A. Bouziad, E. Sukhacheva (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a subset A of the real line , Hattori space H ( A ) is a topological space whose underlying point set is the reals and whose topology is defined as follows: points from A are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on A which are sufficient and necessary for H ( A ) to be respectively almost Čech-complete, Čech-complete, quasicomplete, Čech-analytic and weakly separated...

Spaces with property ( D C ( ω 1 ) )

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

Commentationes Mathematicae Universitatis Carolinae

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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 .