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1.
We show that a completely regular space Y is a p-space (a ?ech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: AY there are a p-embedded subset (resp. a G δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.  相似文献   

2.
A set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F(x) is a nonempty, compact and connected subset of Y for each xX. We denote by L(X), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L(X) endowed with the Vietoris topology. The second author is supported by the SPM fellowship awarded by the Council of Scientific and Industrial Research, India.  相似文献   

3.
The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyYX there exists a Borel setB⊂ℬ(X) such thatBY andm(B) =m(Y), iff:XX is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetCX withm (C) = 1, finitely chaotic with respect to the sequence {m i}, i.e. for any finite subsetA ofC and for any mapF:AX there is a subsequencer i such that limi→∞ f r i(a) =F(a) for anyaA. There are some applications to maps of one dimension. the National Natural Science Foundation of China.  相似文献   

4.
We observe that a separable Banach space X is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if ℒ(X, Y) is not reflexive for reflexive X and Y then ℒ(X 1, Y) is is not reflexive for some X 1X, X 1 having a basis. This work was supported by the grants No. 201/03/0041 and No. 201/04/0090 of the Grant Agency of the Czech Republic and by the grant No. A1019801 of the Academy of Sciences of the Czech Republic.  相似文献   

5.
For metric spaces (X, d x) and (Y, d y) we consider the Hausdorff metric topology on the set (CL(X × Y), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology defined on CL(X × Y). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τ uc on the set F(X, 2Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y),) of closed subsets of Y metrized by the Hausdorff metric . We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank.ubica Holá for suggestions and comments.  相似文献   

6.
Let X and Y be topological spaces such that an arbitrary mapping f: XY for which every preimage f −1 (G) of a set G open in Y is an F σ-set in X can be represented in the form of the pointwise limit of continuous mappings f n : XY. We study the problem of subspaces Z of the space Y for which the mappings f: XZ possess the same property. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1189–1195, September, 2008.  相似文献   

7.
Let X be a topological space, denote iA and cA the interior and the closure of AX, respectively, and let γ = c o i, or = i o c, or = i o c o i, or = c o i o c. A set AX is said to be γ-open [5] iff A ⊂ γ(A). The space X is γ-compact iff each cover of X composed of γ-open sets admits a finite subcover. The purpose of the paper is to investigate some questions concerning γ-compact and related spaces. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

8.
A a set-valued optimization problem min C F(x), xX 0, is considered, where X 0X, X and Y are normed spaces, F: X 0Y is a set-valued function and CY is a closed cone. The solutions of the set-valued problem are defined as pairs (x 0,y 0), y 0F(x 0), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x 0, y 0) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.  相似文献   

9.
Let X be a topological space and (Y,d) be a metric space. If f: XY is a function then there is a function a f : X → [0, ∞] such that f is almost continuous at x if and only if a f (x) = 0. Some properties of this function are investigated. Supported by grant VEGA 2/6087/26 and APVT-51-006904.  相似文献   

10.
We prove that, for an arbitrary Baire space X, a linearly ordered compact set Y, and a separately continuous mapping ƒ: X × Y → R, there exists a G δ-set AX dense in X and such that the function ƒ is jointly continuous at every point of the set A × Y, i.e., any linearly ordered compact set is a co-Namioka space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 1001–1004, July, 2007.  相似文献   

11.
A topological space X is called almost maximal if it is without isolated points and for every xX, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.  相似文献   

12.
We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setFX withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.  相似文献   

13.
A linear continuous nonzero operator G: XY is a Daugavet center if every rank-1 operator T: XY satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X 1⊕F X 2 of two Banach spaces X 1 and X 2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X 1 and X 2 there exists a Daugavet center acting from X 1⊕F X 2, and the class of those F such that for some pair of spaces X 1 and X 2 there is a Daugavet center acting into X 1⊕F X 2. We also present several examples of such Daugavet centers.  相似文献   

14.
Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III 1 or that they are both of type III λ, 0 < λ < 1 and, in the III λ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense G δ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T| X and S| Y, that is ϕ{T i x} = {S i ϕx} for all xX′. Moreover, the Radon-Nikodym derivative ∘ ϕ/dμ is continuous on X′ and, letting S′ = ϕ −1 Sϕ, we have T x = S n(x) x and Sx = T m(x) x where n and m are continuous on X′.  相似文献   

15.
We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x 1 + x 2) ⊂ φ(x 1) + φ(x 2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: XY such that Axφ(x), xX. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.  相似文献   

16.
A topological spaceX is called weakly first countable, if for every pointx there is a countable family {C n x |nω} such thatxC n +1x C n x and such thatUX is open iff for eachxU someC n x is contained inU. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N. N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V. I. Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption b=c, and also by forcing. The second author was supported by the Ben-Gurion University Center for Advanced Studies in Mathematics, Be’er Sheva. The third author was supported by OTKA grant no. 37758 of Hungary.  相似文献   

17.
LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T −1(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX 1 andX 2 such thatX 1 is reflexive and densX 2**=densX**/X.  相似文献   

18.
Let X,i.i.d. and Y1i. i.d. be two sequences of random variables with unknown distribution functions F(x) and G(y) respectively. X, are censored by Y1. In this paper we study the uniform consistency of the Kaplan-Meier estimator under the case ey=sup(t:F(t)<1)>to=sup(t2G(t)<1) The sufficient condition is discussed.  相似文献   

19.
The classical way to study a finite poset (X, ≤ ) using topology is by means of the simplicial complex Δ X of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84–95, 1952), we define the simplicial complexes K X and L X associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex Δ X . We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K X and L X are topologically equivalent to the smaller complexes K X , L X induced by the relation <. More precisely, we prove that K X (resp. L X ) simplicially collapses to K X (resp. L X ). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y.  相似文献   

20.
It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2 X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2 X is homeomorphic to Π2 ifX is a Baire space, and toQ∖Π1 ifX contains a dense absoluteG δ-setGX such that the intersectionGU is connected for any open connectedUX. (Here Π1, Π1X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 35–51, July, 1997. Translated by O. V. Sipacheva  相似文献   

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