Z-拟连续domain上的Scott拓扑和Lawson拓扑

对一般子集系统Z,引入了Z-拟连续domain的概念,证明了Z-完备偏序集P是Z-拟连续的当且仅当P上的Z-Scott拓扑σZ(P)在集包含序下是超连续格;Z-拟连续domain P上的Z-Scott拓扑σZ(P)是Sober的当且仅当σZ(P)具有Rudin性质,P贼予Z-Lawson拓扑λZ(P)是pospace,且若P上的Z-Lawson开上集是Z-Scott开的,Z-Lawson开下集是下拓扑开的,则(P,λZ(P))为严格完全正则序空间.     

Z-拟连续domain上的Scott拓扑和Lawson拓扑

对一般子集系统Z,引入了Z-拟连续domain的概念,证明了Z-完备偏序集P是Z-拟连续的当且仅当P上的Z-Scott拓扑σ_z(P)在集包含序下是超连续格;Z-拟连续domain P上的Z-Scott拓扑σ_z(P)是Sober的当且仅当σ_z(P)具有Rudin性质,P赋予Z-Lawson拓扑λ_z(P)是pospace;且若P上的Z-Lawson开上集是Z-Scott开的,Z-Lawson开下集是下拓扑开的,则(P,λ_z(P))为严格完全正则序空间。     

超连续Domain与拟超连续Domian

讨论了超连续domain与拟超连续domain的相关性质,证明了超连续半格范畴为有性质M*的拟超连续domain范畴的反射子范畴。     

拟Z-连续domain和Z-交连续domain

对一般子集系统Z,引入了Rudin性质、拟Z-连续domain及Z-交连续 domain的概念,讨论了它们的基本性质.特别是Z-连续性、拟Z-连续性、 Z-交连 续性和Z-Lawson拓扑之T2性之间的相互关系. 证明了当子集系统Z满足一定条件 时,拟Z-连续domain P上的Z-way below关系Z具有插入性质, P上的Z-Lawson 拓扑λZ(P)是T2的,且P可用Z-Lawson同态嵌入到某方体之中.文中给出了一个 domain P,其上的Lawson拓扑λ(P)是T2的,但P不是拟连续性domain.     

关于拟连续domain以及函数空间的注记

用伴随的方式给出了拟连续domain以及其Scott闭集格(又叫广义完全分配格)的等价刻画。此外,本文证明了拟连续格关于函数空间是不封闭的,从而不能构成cartesian闭范畴。     

性质M_F与拟连续Domain中的Scott上紧集

对拟连续domain引入了关于基的性质MF,证明了对拟连续domain P,任两个Scott紧上集的交是Scott紧当且仅当P关于某一(任一)基具有性质MF.     

关于交半格同态构成的函数空间与FS-交连续Domain

定义了一类序结构-FS-交连续domain,讨论其相关性质并证明:(1)FS-交连续domain关于由Scott连续且保持非空有限交运算的函数构成的函数空间封闭,以(代数)FS-交连续domain为对象、以Scott连续函数为态射的范畴是Cartesian闭范畴;(2)任意分配可乘的有界完备domain是FS-交连续domain,从而紧连续dcpo的Smyth幂domain是FS-交连续domain.这些结果表明,FS-交连续domain是关于保非空有限交的连续映射构成的函数空间封闭的最恰当序结构.     

关于交半格同态构成的函数空间与FS-交连续Domain

定义了一类序结构—FS-交连续domain,讨论其相关性质并证明:(1)FS-交连续domain关于由Scott连续且保持非空有限交运算的函数构成的函数空间封闭,以(代数)FS-交连续domain为对象、以Scott连续函数为态射的范畴是Cartesian闭范畴;(2)任意分配可乘的有界完备domain是FS-交连续domain,从而紧连续dcpo的Smyth幂domain是FS-交连续domain.这些结果表明,FS-交连续domain是关于保非空有限交的连续映射构成的函数空间封闭的最恰当序结构.     

拟连续格的逆极限

给出以保任意交和定向并为态射的拟连续格范畴的逆极限,讨论函子保广义拟连续格逆极限的条件。     

关于拟连续Domain的一个注记

证明了:(1)具有性质M的dcpo为拟连续domain当且仅当其上的下拓扑开集格在集合包含序下为连续格;(2)对于dcpo L,L为拟连续domain当且仅当ΣL的Hoare空间为局部强紧空间.     

Frink quasicontinuous posets

In this paper, the concept of Frink quasicontinuous posets is introduced. The main results are: (1) a poset is a Frink quasicontinuous poset if and only if its normal completion is a quasicontinuous lattice; (2) a poset is precontinuous if and only if it is Frink quasicontinuous and meet precontinuous; (3) when a Frink quasicontinuous poset satisfies certain conditions, the way below relation has the interpolation property; (4) the category of quasicontinuous lattices with complete homomorphisms is a full reflective subcategory of the category of Frink quasicontinuous posets with cut-stable maps.     

Quasicontinuity of Posets via Scott Topology and Sobrification

In this paper, posets which may not be dcpos are considered. In terms of the Scott topology on posets, the new concept of quasicontinuous posets is introduced. Some properties and characterizations of quasicontinuous posets are examined. The main results are: (1) a poset is quasicontinuous iff the lattice of all Scott open sets is a hypercontinuous lattice; (2) the directed completions of quasicontinuous posets are quasicontinuous domains; (3) A poset is continuous iff it is quasicontinuous and meet continuous, generalizing the relevant result for dcpos. Supported by the NSF of China (10371106, 10410638) and by the Fund (S0667-082) from Nanjing University of Aeronautics and Astronautics.     

QFS-Domains and their Lawson Compactness

In this paper, concepts of quasi-finitely separating maps and quasi-approximate identities are introduced. Based on these concepts, QFS-domains and quasicontinuous maps are defined. Properties and characterizations of QFS-domains are explored. Main results are: (1) finite products, nonempty Scott closed subsets and quasicontinuous projection images of QFS-domains, as well as FS-domains, are all QFS-domains; (2) QFS-domains are compact in the Lawson topology; (3) An L-domain is a QFS-domain iff it is an FS-domain, iff it is compact in the Lawson topology; (4) Bounded complete quasicontinuous domains, in particular quasicontinuous lattices, are all QFS-domains.     

拟连续Domain的拟基及其权

在定向完备偏序集(即dcpo)上引入了拟基的概念,给出了拟基的若干刻画并在此基础上定义了拟连续Domain的权。探讨了拟连续Domain的权与该拟连续Domain上赋予内蕴拓扑时的拓扑空间的权之间的关系。     

Quasicontinuous functions,minimal usco maps and topology of pointwise convergence

In [HOLá, Ľ.—HOLY, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math.] a complete answer is given, for a Baire space X, to the question of when the pointwise limit of a sequence of real-valued quasicontinuous functions defined on X is quasicontinuous. In [HOLá, Ľ.—HOLY, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562], a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is proved. Continuing these results, we study closed and compact subsets of the space of quasicontinuous functions and minimal USCO maps equipped with the topology of pointwise convergence. We also study conditions under which the closure of the graph of a set-valued mapping which is the pointwise limit of a net of set-valued mappings, is a minimal USCO map.     

拓扑空间范畴、拓扑FUZZ范畴与拓扑分子格范畴间的反射与余反射

从整体角度出发，证明了拓扑空间范畴Ｔｏｐ分别是拓扑Ｆｕｚｚ范畴ＴｏｐＦｕｚ与拓扑分子格范畴ＴＭＬ的反射与余反射满子范畴，ＴｏｐＦｕｚ是ＴＭＬ的反射与余反射（非满）子范畴．     

Quotients of internally quasicontinuous functions<Superscript>*</Superscript>

In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family.