关于具有σ几乎局部有限基的拓扑空间

In 1983, ML Ito and K.Tamano introduced the notion of almost locally finiteness and investigated some properties of the class of spaces with a σ almost locally finite base. In this paper, some more results about such spaces are given . The main results are as follows:( 1 ) The image of a space with a σ -almost locally finite base under the finite to one closed map has a σ-almost locally finite base;( 2 ) The locally finite sum theorem holds for such spaces;( 3 ) A space X has a σ almost locally finite base if and only if X is a paracompact σ space and every closed subset of X has an almost locally finite open neighborhood base; (4)The monotonically normal space with an M-structure has a a σalmost localiy finite base.     

Fuzzy蕴涵代数的素MP-滤子空间(英文)

In this paper,the topological space(PF_MP（X）,T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T₀-space if X with condition（P）.Secondly,we restricted T to the set of all maximal MP-filters MF_MP（X） of X and concluded that(PF_MP（X）,T |_PFMP（X） )is a compact T₂ space if X with conditions（P） and（S）.     

On P-sober Spaces

In this paper,we focus on p-sober spaces and prove that(1) the To space X is p-sober if and only if the Smyth power space of X is p-sober;(2) the space X has a p-sober dcpo model if and only if X is T₁ and p-sober;(3) every non-p-sober T₀ space does not have a p-sobrification;(4) the T₀ space X is sober if and only if X is p-sober and PD.     

Characterizations and Extensions of Lipschitz-α Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Inverse limits of hereditarily almost expandable class

In this paper, the hereditarily almost expandability of inverse limits is investigated, with two results obtained. Let X be the inverse limit space of an inverse system （Xα, π^αβ, ∧}. （1） Suppose X is hereditarily κ-metacompact, if each Xα is hereditarily pointwise collectionwise normal （almost θ-expandable, almost discrete θ-expandable）, then so is X; （2） Suppose X is hereditarily κ-σ-metacompact, if each Xα is hereditarily almost σ-expandable （almost discrete σ-expandable = σ-pointwise collectionwise normal）, then so is X.     

Ideals in the Roe Algebras of Discrete Metric Spaces with Coefficients in B（H）

The notion of an ideal family of weighted subspaces of a discrete metric space X with bounded geometry is introduced. It is shown that, if X has Yu＇s property A, the ideal structure of the Roe algebra of X with coefficients in B（H） is completely characterized by the ideal families of weighted subspaces of X, where B（H） denotes the C＊-algebra of bounded linear operators on a separable Hilbert space H.     

Division and k-th root theorems for Q-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f:K→X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X×[0,1] are Q-manifolds as well. We construct a countable familyχof spaces with DDP and cd-AP such that no space X∈χis homeomorphic to the Hilbert cube Q whereas the product X×Y of any different spaces X, Y∈χis homeomorphic to Q. We also show that no uncountable familyχwith such properties exists.     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T:X∗→Y can be uniformly approximated by finite rank operators from X⊗Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T:X∗→Y can be approximated in the strong operator topology by operators of norm ?‖T‖ from X⊗Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.     

On nagata's problem for paracompact σ-metric spaces

Let (A) be the characterization of dimension as follows: Ind X?n if and only if X has a σ-closure-preserving base $\text{W}$ such that Ind B(W)?n?1 for every W?$\text{W}$. The validity of (A) is proved for spaces X such that(i) X is a paracompact σ-metric space with a scale {X_i} such that each X_i has a uniformly approaching anti-cover, or(ii) X is a subspace of the product ΠX_i of countably many L-spaces X_i, the notion of which is due to K. Nagami.(i) and (ii) are the partial answers to Nagata's problem wheter (A) holds or not for every M₁-space X.     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

More stratifiable function spaces

If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M₁.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

A discrete homotopy theory for binary reflexive structures

We present a simple combinatorial construction of a sequence of functors σ_k from the category of pointed binary reflexive structures to the category of groups. We prove that if the relational structure is a poset P then the groups are (naturally) isomorphic to the homotopy groups of P when viewed as a topological space with the topology of ideals, or equivalently, to the homotopy groups of the simplicial complex associated to P. We deduce that the group σ_k(X,x₀) of the pointed structure (X,x₀) is (naturally) isomorphic to the kth homotopy group of the simplicial complex of simplices of X, i.e. those subsets of X which are the homomorphic image of a finite totally ordered set.     

Detecting codimension one manifold factors with 0-stitched disks

The 0-stitched disks property is introduced and shown to detect codimension one manifold factors of dimension n?4. It is shown that if a space X is an ANR and has the 0-stitched disks property, then X has the disjoint homotopies property. It follows that if a space X is a resolvable generalized manifold of dimension n?4 with the 0-stitched disks property, then X is a codimension one manifold factor. Whether or not the 0-stitched disks property is equivalent to the disjoint homotopies property remains an open question.     

Sequences of semicontinuous functions accompanying continuous functions

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.