Locale同伦理论

本文引入了locale连续映射同伦的概念,建立了locale同伦范畴,构造性地证明了任一locale连续映射都同伦等价于一个locale包含映射。通过引入locale H群的概念(它是locale群概念的自然推广),建立了locale同伦范畴到群同态范畴的一个反变函子。特别地,我们建立了locale同伦群范畴上的基本群函子,证明了locale L上以p为基点的基本群同构于L的谱空间pt(L)上以p为基点的基本群。因此,基本群函子是locale范畴中的一个同伦不变量。     

极大点空间的若干特征定理

本文利用极大点空间的等价刻划证明了极大点空间的某些子空间、不交和、 乘积空间、逆序列的逆极限、具有可数基的局部紧的Hausdorff空间是极大点空间,还 给出了具有可数基的局部紧的Hausdorff空间的Domain hull.     

Resolution of the uniform lower bound problem in constructive analysis

In a previous paper we constructed a full and faithful functor ?? from the category of locally compact metric spaces to the category of formal topologies (representations of locales). Here we show that for a real‐valued continuous function f, ??(f) factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than point‐wise positivity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

弱空间式Locale

空间式locale范畴SLoc是locale范畴Loc的余反射满子范畴,但对locale乘积不封闭．本文引入弱空间式locale,证明弱空间式locale范畴WSloc为范畴Loc的余反射满子范畴,且对locale秉积封闭．还证明了一个locale A是空间式的当且仅当它的枝映射localeN(A)是弱空间式的;一个空问式locale的每一个子locale都是空间式的当且仅当它的每一个子locale是弱空间式的．最后,证明了弱空间式性在定向函子下保持不变．     

Lax proper maps of locales

We give a proof of localic Priestley duality. Our approach is based on lax proper maps of locales, which provide a vehicle for presenting the Priestley version of full Stone duality constructively and preserve spatial intuitions.     

Quotient Maps of Locales

We consider regular epimorphisms in the category Loc of locales. Closed surjections with subfit domain are regular epimorphisms. However, there exists a closed surjection which is the composite of two regular epimorphisms without being regular epic itself. This example answers both of the following questions in the negative: Y. Li"s question of whether weak quotient maps are necessarily regular epimorphisms, and P. Johnstone"s related question of whether regular epimorphisms compose. It follows that not all extremal epimorphisms in Loc are regular. The weak quotient maps of Y. Li and the equationally closed subframes of A. Pultr and A. Tozzi are shown to be dual notions. We also give a new characterization of regular epimorphisms in Loc.     

On Uniformly Locally Compact Quasi-Uniform Hyperspaces

We characterize those Tychonoff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family of nonempty compact subsets of X. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space Xis uniformly locally compact on if and only if Xis paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is -compact if and only if its (lower) semi-continuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on is obtained.     

Locale范畴中的零维性

本文讨论ｌｏｃａｌｅ的零维性质，主要结果有：（１）给出ｌｏｃａｌｅＡ的核映射（ｎｕｃｌｅｕｓ）构成的ｌｏｃａｌｅＮ（Ａ）中上确界的点式刻划，并得到了Ｎ（Ａ）的紧性与Ａ的紧性之间的关系；（２）给出零维ｌｏｃａｌｅ与ｃｏｈｅｒｅｎｔｌｏｃａｌｅ之间的关系，以及零维ｌｏｃａｌｅ的紧零维反射；（３）给出零维ｌｏｃａｌｅ范畴在ｌｏｃａｌｅ范畴中的刻划．     

k-COVERS AND CERTAIN QUOTIENT IMAGES OF PARACOMPACT LOCALLY COMPACT SPACES

We continue to discuss some properties of certain k-covers. As applications, we give some characterizations of certain quotient images of paracompact locally compact spaces.