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For a space X denote by Cb(X) the Banach algebra of all continuous bounded scalar-valued functions on X and denote by C0(X) the set of all elements in Cb(X) which vanish at infinity. 相似文献
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Let X be a compact abelian group. A subgroup H of X is called characterized if there exists a sequence u=(un) of characters of X such that H=su(X), where su(X):={x∈X:(un,x)→0 in T}. Every characterized subgroup is an Fσδ-subgroup of X . We show that every Gδ-subgroup of X is characterized. On the other hand, X has non-characterized Fσ-subgroups. 相似文献
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In this paper, we introduce the metric dG on a G -metric space (X,G) and use this notion to show that many contraction conditions for maps on the G -metric space (X,G) reduce to certain contraction conditions for maps on the metric space (X,dG). As applications, the proofs of many fixed point theorems for maps on the G -metric space (X,G) may be simplified, and many fixed point theorems for maps on the G -metric space (X,G) are direct consequences of preceding results for maps on the metric space (X,dG). 相似文献
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The main result of the paper says that if X is a paracompact GO-space, meaning a subspace of a linearly ordered space and M a paracompact space satisfying the first axiom of countability such that X can be embedded in Mω1 then the product X×Y is paracompact for every paracompact space Y if and only if the first player of the G(DC,X) game, introduced by Telgarsky has a winning strategy. In particular we obtain that if X is paracompact GO-space of weight not greater than ω1 then the product X×Y is paracompact for every paracompact space Y if and only if the first player of the G(DC,X) game has a winning strategy. 相似文献
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A weak selection on an infinite set X is a function σ:[X]2→X such that σ({x,y})∈{x,y} for each {x,y}∈[X]2. A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X]2 and the topology on X . We study some topological consequences from the existence of a continuous weak selection on the product X×Y for the following particular cases:
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- Both X and Y are spaces with one non-isolated point. 相似文献
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Given a finite, connected 2-complex X such that b2(X)?1 we establish two existence results for representations of the fundamental group of X into compact connected Lie groups G , with prescribed values on certain loops. If b2(X)=1 we assume G=SO(3) and that the cup product on H1(X;Q) is non-zero. 相似文献
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For a Tychonoff space X , we denote by Cp(X) and Cc(X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X in terms of Cp(X), we extend Okunev?s results by showing that if there exists a surjection from Cp(X) onto Cp(Y) (resp. from Lp(X) onto Lp(Y)) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from Cc(X) onto Cc(Y), then Y is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by ?p-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and ?p-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented. 相似文献
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We introduce the monotone Sokolov property and show that it is dual to monotone retractability in the sense that X is monotonically retractable if and only if Cp(X) is monotonically Sokolov. Besides, a space X is monotonically Sokolov if and only if Cp(X) is monotonically retractable. Monotone retractability and monotone Sokolov property are shown to be preserved by R-quotient images and Fσ-subspaces. Furthermore, every monotonically retractable space is Sokolov so it is collectionwise normal and has countable extent. We also establish that if X and Cp(X) are Lindelöf Σ-spaces then they are both monotonically retractable and have the monotone Sokolov property. An example is given of a space X such that Cp(X) has the Lindelöf Σ-property but neither X nor Cp(X) is monotonically retractable. We also establish that every Lindelöf Σ-space with a unique non-isolated point is monotonically retractable. On the other hand, each Lindelöf space with a unique non-isolated point is monotonically Sokolov. 相似文献
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Let K be an algebraically closed field of characteristic 0 and let Mn(K), n?3, be the matrix ring over K . We will show that the image of any multilinear polynomial in four variables evaluated on Mn(K) contains all matrices of trace 0. 相似文献
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Roe algebras are C?-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d). In the special case that X=Γ is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (Γ,d) is isomorphic to the crossed product C?-algebra l∞(Γ)?rΓ. 相似文献
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We prove that for any free ergodic nonsingular nonamenable action Γ?(X,μ) of all Γ in a large class of groups including all hyperbolic groups, the associated group measure space von Neumann algebra L∞(X)?Γ has L∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in Popa and Vaes (in press) [38]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products M1?BM2 over a subalgebra B of type I. 相似文献
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Let X be a Banach space and L the generator of the evolution semigroup associated with the τ -periodic evolutionary process {U(t,s)}t≥s on the space Pτ(X) of all τ-periodic continuous X -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?) under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0). The proof is based on a new decomposition of the space Pτ(X) by constructing a right inverse of L. 相似文献