共查询到20条相似文献,搜索用时 31 毫秒
1.
Horst Herrlich 《Topology and its Applications》2009,156(11):1962-1965
For topological products the concept of canonical subbase-compactness is introduced, and the question analyzed under what conditions such products are canonically subbase-compact in ZF-set theory.Results: (1) Products of finite spaces are canonically subbase-compact iff AC(fin), the axiom of choice for finite sets, holds.(2) Products of n-element spaces are canonically subbase-compact iff AC(<n), the axiom of choice for sets with less than n elements, holds.(3) Products of compact spaces are canonically subbase-compact iff AC, the axiom of choice, holds.(4) All powers XI of a compact space X are canonically subbase compact iff X is a Loeb-space.These results imply that in ZF the implications
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The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M. 相似文献
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Horst Herrlich 《Topology and its Applications》2011,158(17):2279-2286
Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:
- (1)
- For every family(Ai)i∈Iof sets there exists a family(Ti)i∈Isuch that for everyi∈I(Ai,Ti)is a compactT2space.
- (2)
- For every family(Ai)i∈Iof sets there exists a family(Ti)i∈Isuch that for everyi∈I(Ai,Ti)is a compact, scattered, T2space.
- (3)
- For every set X, every compactR1topology (itsT0-reflection isT2) on X can be enlarged to a compactT2topology.
- (a)
- (1) implies every infinite set can be split into two infinite sets.
- (b)
- (2) iff AC.
- (c)
- (3) and “there exists a free ultrafilter” iff AC.
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Much of General Topology addresses this issue: Given a function f∈C(Y,Z) with Y⊆Y′ and Z⊆Z′, find , or at least , such that ; sometimes Z=Z′ is demanded. In this spirit the authors prove several quite general theorems in the context Y′=κ(XI)=∏i∈IXi in the κ-box topology (that is, with basic open sets of the form ∏i∈IUi with Ui open in Xi and with Ui≠Xi for <κ-many i∈I). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this.
Theorem.
Letω?κ?α (that means: κ<α, and[β<αandλ<κ]⇒βλ<α) with α regular,be a set of non-empty spaces with eachd(Xi)<α,π[Y]=XJfor each non-emptyJ⊆Isuch that|J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets ofZ×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for everyf∈C(Y,Z)there isJ∈[I]<αsuch thatf(x)=f(y)wheneverxJ=yJ, and (c) every such f extends to. 相似文献
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Jan Paseka 《Topology and its Applications》2008,155(4):308-317
We begin with the notion of K-flat projectivity. For each sup-algebra L we then introduce a binary relation L? on it. The K-flat projective sup-algebras are exactly such sup-algebras with each element a approximated by the element x, xL?a and the relation L? being stable with respect to the operations on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective sup-algebras as such sup-algebras having a coalgebra structure for the K-comonad. 相似文献
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A space X is said to have property (USC) (resp. (LSC)) if whenever is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [0,1] converging pointwise to the constant function 0 with the value 0, there is a sequence of continuous functions from X into [0,1] such that fn?gn (n∈ω) and converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S1(Γ,Γ) and wQN. 相似文献
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Sibe Mardeši? 《Topology and its Applications》2009,156(14):2326-2345
In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top). 相似文献
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Let (X1,X2,…,Xn) and (Y1,Y2,…,Yn) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ1,λ2,…,λn), (μ1,μ2,…,μn), respectively. Let X()=(X(1),X(2),…,X(n)), Y()=(Y(1),Y(2),…,Y(n)) be the order statistics of (X1,X2,…,Xn) and (Y1,Y2,…,Yn). Then (λ1,λ2,…,λn) majorizes (μ1,μ2,…,μn) implies that X() is stochastically larger than Y(). However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results. 相似文献
14.
Ziqin FengPaul Gartside 《Topology and its Applications》2011,158(9):1124-1130
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J.A. Brown 《Topology and its Applications》2008,155(4):190-200
Assume CH. Let I be any index set, and let Xi, for i∈I, be a completely regular ccc topological space of weight ω2. If X=∏i∈IXi is ccc and non-pseudocompact, then X has remote points. 相似文献
16.
Omar Hirzallah 《Linear algebra and its applications》2007,424(1):71-82
We prove several singular value inequalities and norm inequalities involving sums and direct sums of Hilbert space operators. It is shown, among other inequalities, that if X and Y are compact operators, then the singular values of are dominated by those of X ⊕ Y. Applications of these inequalities are also given. 相似文献
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In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T0, T1, and T2 preordered sets and show that each of the full subcategories of each of T0, T1, T2 preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord. 相似文献
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Kazushi Yoshitomi 《Indagationes Mathematicae》2005,16(2):289-299
Let q ∈ {2, 3} and let 0 = s0 < s1 < … < sq = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set lj = sj − sj−1 for j ∈ 1, q. Given (p1,, pq) ∈ Rq, let b: Z → R be a periodic function of period T such that b(·) = pj on sj−1 + 1, sj for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l2(Z). By [λ2j , λ2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that pm ≠ pn for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1. 相似文献
19.
Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron II
Sibe Mardeši? 《Topology and its Applications》2008,155(15):1708-1719
In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In a subsequent paper, published in 2007, the author proved that R(X,K) is a covariant functor in the first variable. In the present paper it is proved that R(X,K) is a covariant functor also in the second variable. 相似文献
20.
Wasin So 《Linear algebra and its applications》2010,432(9):2163-471
The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X+Y=Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities. 相似文献