关于Locale范畴反射子范畴的注解

利用Locale中的完全正则元和零维元构造性地给出了任意Locale的完全正则反射以及零维反射的描述,并且对于满足‘(?)’关系插入性的Locale,特别地,对正规Locale,证明了全体正则元构成的Locale是其正则反射.进而,利用平稳(flat)子Locale的扩张引理给出了Locale的紧完全正则反射,紧零维反射以及紧正则反射的构造性描述.     

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．     

Locale范畴的反射子范畴

本文研究ｌｏｃａｌｅ范畴的反射子范畴，给出反射子范畴的刻划定理，从一般的ｌｏｃａｌｅ出发，完全构造性地给出了ｌｏｃａｌｅ的正则反射、完全正则反射和零维反射的构造．     

Locale的仿紧完全正则反射

1972年, Isbell利用locale中由正规覆盖构成的一致结构的完备化, 证明了仿紧完全正则locale范畴是locale范畴的满反射子范畴. 本文通过在locale的补零元理想格上做核映射的方法, 给出locale的仿紧完全正则反射的明确构造, 并证明locale的仿紧完全正则反射是locale的 Stone-Čech紧化的子locale.     

一致Locale的乘积

本文利用Ｌｏｃａｌｅ上的收敛结构研究一致Ｌｏｃａｌｅ的乘积结构，证明了一致Ｌｏ ｃａｌｅ的完备性关于Ｌｏｃａｌｅ积和弱积封闭．     

Locale的正则紧反射

ｌｏｃａｌｅ的正则紧反射函子的构造的明确描述问题是由ＢａｎａｓｃｈｅｗｓｋｉＢ．和ＭｕｌｖｅｙＣ．Ｊ．于１９８０年提出的,十多年来一直没有进展。本文通过在ｌｏｃａｌｅ上引入一种二元关系,给出了ｌｏｃａｌｅ的正则紧反射函子的构造性描述。     

Locale的函数空间

构造性地定义了Locale间的Isbell,紧开及点态函数空间,研究了其良好的范畴性质(Proper和Admissible性质)及分离性质.重新得到了有指数Locale的结构.这些工作是寻找Locale范畴中的方便范畴的基础。     

Locale的函数空间

构造性地定义了Locale间的Isbell,紧开及点态函数空间,研究了其良好的范畴性质(Proper和Admissible性质)及分离性质.重新得到了有指数Locale的结构.这些工作是寻找Locale范畴中的方便范畴的基础.     

关于3—正则子图问题的一个注记

这篇注记证明判断一个图是否有３－正则子图的问题，即使对于节点次不超过４的平面力，仍然是ＮＰ－完全的，而且，此结果是最好的可能。     

Isocompactness in the Category of Locales

We study isocompactness in Loc defined, exactly as in Top, by requiring that every countably compact closed sublocale be compact. This is a genuine extension of the same-named topological concept since every Boolean (or, even more emphatically, every paracompact) locale is isocompact. A slightly stronger variant is defined by decreeing that the closure of every complemented countably compact sublocale be compact. Dropping the adjective “complemented” yields a formally even stronger property, which we show to be preserved by finite products. Metrizable locales (or, more generally, perfectly normal locales) do not distinguish between the three variants of isocompactness. Each of the stronger variants of isocompactness travels across a proper map of locales, and in the opposite direction if the map is a surjection in Loc.     

关于Locale Wallman紧化的一点注记

本文证明了在正规locale范畴中,Banaschewski-Mulvey形式的紧正则反射与Johnstone形式的Wallman紧化一致,从而推广了Johnstone在文献[4]中的主要结果.     

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.     

Locale上的收敛结构

本文从点式处理角度，在Ｌｏｃａｌｅ上成功地建立了收敛结构。利用它研究了一致Ｌｏｃａｌｅ的完备性、刻划紧性，、界定了Ｌｏｃａｌｅ上一种满意的Ｈａｕｓｄｏｒｆｆ分离性，     

A Note on Proper Maps of Locales

Let X and Y be completely regular locales. We show that the properness of a localic map f: X → Y can be characterized in terms of extension between compactifications.     

A Convenient Subcategory of Tych

A map f:XY between Hausdorff topological spaces is k-continuous if its restriction f| _K to every compact subspace K of X is continuous. X is called a k _R -space if every k-continuous function from X to a Tychonoff space is continuous. In this paper we investigate the category of Tychonoff k _R -spaces, and show that it is Cartesian closed (thus convenient in the sense of Wyler).