Remarks on Dense Subspaces

Some constructions of spaces with/without dense subspaces satisfying stronger separation axioms are presented.     

On a Property of Urysohn's Universal Metric Space

Mathematical Notes -     

关于Bloch空间的子空间

本文给出了Ｂｌｏｃｈ空间一类子空间的包含关系,并利用Ｈａｄａｍａｒｄ缺项级数证明了这些关系是最好的．所获得的结果包含了Ｂｅｓｏｖ空间和Ｂｌｏｃｈ空间的一些已知结论．     

关于满足强分离开集条件的自相似集的Hausdorff测度

设E是Rn中由相似压缩S1,S2,…,Sm所确定的满足开集条件的自相似集,其Hausdorff维数为s,其s-维Hausdorff测度记为Hs(E).利用部分估计原理得到了本文的主要结果:若E满足强分离开集条件,则在E中存在一个压缩拷贝串序列{Ui}和紧集U(|U|>0),使得Hs(U)等于|U|s,并且{Ui}按Hausdorff度量收敛到U,进而证明了由U可以构造一个数列,使得该数列正好收敛到Hs(E);另外,引入了自相似集的相似压缩不动点,得到了等式Hs(E∩U)=|U|s 成立的一个必要条件.     

*-仿紧性与分离公理

对L-fuzzy子集的*-仿紧性进行修正,重证两个定理,另外证明*-仿紧的Hausdorff空间是S^*正则的与S^*正规的。     

Algebras and Spaces of Dense Constancies

A DC-space (or space of dense constancies) is a Tychonoff space X such that for each f C(X) there is a family of open sets {U _i : i I}, the union of which is dense in X, such that f, restricted to each U _i , is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean f-algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions are dense), and it is shown that all metrizable spaces have this property.     

On Subspaces and Mappings of Near-vector Spaces

We investigate subspaces and homogeneous and linear mappings of near-vector spaces.     

L-fuzzy拓扑空间中关于子基的分离性

在L-fuzzy拓扑空间中引入了关于子基的分离性公理,给出了它们的特征刻画,研究了它们的一系列基本性质,如可遗传性、可积性、L-fuzzy同胚不变性等,并得到了这类新分离性与T分离性是等价的.     

On the Weight of Nowhere Dense Subsets in Compact Spaces

We study a new cardinal-valued invariant ndw(X) (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points.     

Subspaces of Normed Riesz Spaces

It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm p satisfies p(x)p(y) for all x and y with -yxy. A similar characterization of the subspaces of M-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the r-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

On the dynamics of a family of generated renormalization transformations

We study the family of renormalization transformations of the generalized d  -dimensional diamond hierarchical Potts model in statistical mechanic and prove that their Julia sets and non-escaping loci are always connected, where d?2$d?2$. In particular, we prove that their Julia sets can never be a Sierpiński carpet if the parameter is real. We show that the Julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models. Moreover, the asymptotic formula of the Hausdorff dimension of the Julia set is calculated as the parameter tends to infinity.     

Sobriety in Terms of Nets

Sobriety is a subtle notion of completeness for topological spaces: A space is sober if it may be reconstructed from the lattice of its open subsets. The usual criterion to check sobriety involves either irreducible closed subsets or completely prime filters of open sets. This paper provides an alternative possibility, thus trying to make sobriety easier to understand. We define the notion of observative net, which, together with an appropriate convergence notion, characterizes sobriety. As the filter approach does not involve just usual (topological) convergence, this is not an instance of the classical net-filter translation in general topology.     

关于序列次可分解算子的不变子空间

得到了关于序列次可分解算子的一个不变子空间定理，推广了H.Mohebi和M.Rajiabalipour在1994年得到的一个不变子空间定理，并且举例说明存在l2上的有界线性算子T。它有无穷多个变子空间，但是它的不变子空间格Lat(T)不丰富。     

自相似集的质量分布原理与Hausdorff测度及其应用

本文提出了满足开集条件的自相似集的质量分布原理.作为应用,得到了计算一类满足开集条件的自相似集的Hausdorff测度的准确值的方法,并举例说明了此方法对于计算一类满足开集条件的自相似集的Hausdorff测度的准确值是行之有效的.     

关于自相似集的Hausdorff测度的一个判据及其应用

讨论了满足开集条件的自相似集。对于此类分形，用自然覆盖类估计它的Hausdorff测度只能得到一个上限，因而如何判断某一个上限就是它的Hausdorff测度的准确值是一个重要的问题。本文给出了一个判据。作为应用，统一处理了一类自相似集，得到了平面上的一个Cantor集－Cantor尘的Hausdorff测度的准确值，并重新计算了直线上的Cantor集以及一个Sierpinski地毯的Hausdorff测度。     

L-fuzzy拓扑空间中的包含式正则分离性

本文首先在一般L-fuzzy拓扑空间中引入了包含式正则分离性的概念.其次将包含式正则分离性与点式正则分离性及正则分离性作了比较,讨论了它们之间的相互蕴含关系.最后说明了包含式正则分离性与包含式正规分离性及包含式完全正则分离性之间的协调性.     

On countable choice and sequential spaces

Under the axiom of choice, every first countable space is a Fréchet‐Urysohn space. Although, in its absence even ? may fail to be a sequential space. Our goal in this paper is to discuss under which set‐theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ?, are classes of Fréchet‐Urysohn or sequential spaces. In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion. Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ? if and only if the axiom of countable choice holds for families of subsets of ?, and every metric space has a unique ‐completion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hausdorff measure of sets of finite type of one-sided symbolic space

Some estimation formulae and a computation formula for the Hausdorff measure of the sets of finite type of the one-sided symbolic space are given. Project partially supported by the Natural Science Foundation of Guangdong Province and the Foundation of Advanced Research of Zhongshan University.