共查询到20条相似文献,搜索用时 15 毫秒
1.
Kyriakos Keremedis 《Mathematical Logic Quarterly》2012,58(3):130-138
Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:
- (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
- (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
- (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
- (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.
2.
The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters
Eric J. Hall Kyriakos Keremedis Eleftherios Tachtsis 《Mathematical Logic Quarterly》2013,59(4-5):258-267
Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show:
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3.
Kyriakos Keremedis 《Mathematical Logic Quarterly》2010,56(5):475-487
Let {(Xi, Ti): i ∈I } be a family of compact spaces and let X be their Tychonoff product. ??(X) denotes the family of all basic non‐trivial closed subsets of X and ??R(X) denotes the family of all closed subsets H = V × ΠXi of X, where V is a non‐trivial closed subset of ΠXi and QH is a finite non‐empty subset of I. We show: (i) Every filterbase ?? ? ??R(X) extends to a ??R(X)‐ultrafilter ? if and only if every family H ? ??(X) with the finite intersection property (fip for abbreviation) extends to a maximal ??(X) family F with the fip. (ii) The proposition “if every filterbase ?? ? ??R(X) extends to a ??R(X)‐ultrafilter ?, then X is compact” is not provable in ZF. (iii) The statement “for every family {(Xi, Ti): i ∈ I } of compact spaces, every filterbase ?? ? ??R(Y), Y = Πi ∈IYi, extends to a ??R(Y)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {(Xi, Ti): i ∈ ω } of compact spaces, every countable filterbase ?? ? ??R(X), X = Πi ∈ωXi, extends to a ??R(X)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem restricted to countable families. (v) The countable Axiom of Choice is equivalent to the proposition “for every family {(Xi, Ti): i ∈ ω } of compact topological spaces, every countable family ?? ? ??(X) with the fip extends to a maximal ??(X) family ? with the fip” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
Omar De la Cruz Eric Hall Paul Howard Kyriakos Keremedis Eleftherios Tachtsis 《Mathematical Logic Quarterly》2005,51(6):598-609
We investigate, within the framework of Zermelo‐Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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6.
Horst Herrlich 《Mathematical Logic Quarterly》2003,49(3):323-324
It is shown that in ZF set theory the axiom of choice holds iff every non empty topological space has a maximal closed filter. 相似文献
7.
Paul Howard 《Mathematical Logic Quarterly》1993,39(1):353-356
The deductive strengths of three variations of Rado's selection lemma are studied in set theory without the axiom of choice. Two are shown to be equivalent to Rado's lemma and the third to the Boolean prime ideal theorem. MSC: 03E25, 04A25, 06E05. 相似文献
8.
Kyriakos Keremedis Eleftherios Tachtsis 《Proceedings of the American Mathematical Society》2005,133(12):3691-3701
In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space has a choice function, then so does the family of all non-empty, open subsets of . In addition, we establish that the converse is not provable in ZF.
We also show that the statement ``every subspace of the real line with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form ``every continuum sized family of non-empty subsets of reals has a choice function".
9.
Yasuo Kanai 《Mathematical Logic Quarterly》2002,48(3):413-426
In this article, we shall show the generalized notions of distributivity of Boolean algebras have essential relations with several axioms and properties of set theory, say the Axiom of Choice, the Axiom of Dependence Choice, the Prime Ideal Theorems, Martin's axioms, Lebesgue measurability and so on. 相似文献
10.
K.L. Kozlov 《Topology and its Applications》2010,157(4):698-707
The approach to the problem of the distribution of the functors of the Stone-?ech compactification, the Hewitt realcompactification or the Dieudonné completion with the operation of taking products is discussed using uniform structures on products. In particular, the role of different rectangular conditions is shown. Relative analogues of this question and new examples of (strongly) rectangular products are presented. Characterizations of bounded rectangular subsets of the product are given. 相似文献
11.
Working within a plain texture (S,S), the authors construct a completion of a dicovering uniformity υ on (S,S) in terms of prime S-filters. In case υ is separated, a separated completion is then obtained using the T0-quotient, and it is shown that this construction produces a reflector. For a totally bounded di-uniformity it is verified that these constructions lead to dicompactifications of the uniform ditopology. A condition is given under which complementation is preserved on passing to these completions, and an example on the real texture (R,R,ρ) is presented. 相似文献
12.
We extend earlier work [M.R. Burke, N.D. Macheras, K. Musia?, W. Strauss, Category product densities and liftings, Topology Appl. 153 (2006) 1164-1191] of the authors on the existence of category liftings in the product of two topological spaces X and Y such that X×Y is a Baire space. For given densities ρ, σ on X and Y, respectively, we introduce two ‘Fubini type’ products ρ⊙σ and ρ?σ on X×Y. We present a necessary and sufficient condition for ρ⊙σ to be a density. Provided (X,Y) and (Y,X) have the Kuratowski-Ulam property, we prove for given category liftings ρ, σ on the factors the existence of a category lifting π on the product, dominating the density ρ?σ and such that
13.
M. A. Bastos Yu. I. Karlovich A. F. dos Santos 《Integral Equations and Operator Theory》2002,42(1):22-56
The invertibility of convolution type operators on unions of intervals is studied. Sufficient conditions of invertibility for some classes of these operators are established. Solvability results forn-term corona problems are obtained using two different approaches: one involving reduction ton–1 Riemann-Hilbert problems in two variables and another involving reduction to two-term corona problems. The invertibility of the convolution operators on a union of intervals is also related to the invertibility of associated convolution operators on single intervals. Formulas for the inverse operators are given.Work partially supported by F. C. T. (Portugal) Project Praxis XXI/2/2.1/MAT/441/94 and by CONACYT (México) grant, Cátedra Patrimonial, No. 990017-EX. 相似文献
14.
M.G. Tkačenko 《Topology and its Applications》1983,15(1):93-98
We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Πα?AXα has a countable o-tightness and Y is a subset of X such that πB(Y) = Πα?BXα for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding. 相似文献
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16.
The blow‐up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow 下载免费PDF全文
In this paper, we prove two blow‐up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (?Δ)αu in the space and one for the inviscid Oldroyd model. Assume that (u(t,x),F(t,x)) is a smooth solution to the generalized Oldroyd model in [0,T); then, the solution (u(t,x),F(t,x)) does not develop singularity until t = T provided . For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u,F) can be extended beyond T if , which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
17.
Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module $\mathcal {M}$ are given and the respective lattices $\mathrm{Cong}\mathcal {M}$ and $\mathrm{Ide}\mathcal {M}$ are presented. A characterization of the simple bijective Boolean modules is achieved differing from that given by Brink in a homogeneous approach. We construct the smallest and the greatest modular congruence having the same Boolean part. The same is established for modular ideals. The notions of kernel of a modular congruence and the congruence induced by a modular ideal are introduced to describe an isomorphism between $\mathrm{Cong}\mathcal {M}$ and $\mathrm{Ide}\mathcal {M}$. This isomorphism leads us to conclude that the class of the Boolean module is ideal determined. 相似文献
18.
Peter Zizler Keith F. Taylor Shigeru Arimoto 《Integral Equations and Operator Theory》1997,28(2):245-250
We show that ifT(F) is a selfadjoint block Toeplitz operator generated by a trigonometric matrix polynomialF, then the spectrum ofT(F) as well as the limiting set (F) of the eigenvalues of the truncationsT
n
(F) is the union of a finite collection of segments (the spectral range ofF) and at most a finite set of points for which we give an upper bound. 相似文献
19.
Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L2 restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived. 相似文献
20.
This paper is concerned with large- error estimates concerning convergence in distribution as well as norm convergence for Banach space-valued martingale difference sequences. Indeed, two general limit theorems equipped with rates of convergence for such difference sequences are established. Applications of these lead to the central limit theorem and the weak law of large numbers with rates for Banach space-valued martingales. 相似文献