交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

C_Z-偏序集

我们将强Z-连续偏序集推广到了CZ-偏序集,并讨论了CZ-偏序集和强Z-连续偏序集之间的关系。同时我们定义了CZ-偏序集上的CZ-连续映射,得到CZ-偏序集在该映射下的像集仍是CZ-偏序集。最后,我们讨论了CZ-偏序集上的基及其相关性质。     

强Raney偏序集

引入强Raney偏序集的概念,讨论了强Raney偏序集的一些性质,证明了强Raney偏序集为超代数偏序集,定向完备的偏序集为强Raney偏序集当且仅当它既是Raney偏序集也是A-偏序集.     

L-连续偏序集的M性质与有限分离性

讨论了L-连续偏序集的M性质与有限分离性质之间的关系,主要结果: (1)若P为L-连续偏序集,则P是有限卜集生成,而且满足M性质当且仅当它的定向完备化为FS-domain(有限分离的domain);(2)若P为相容L-domain,则P是有限上集生成,而且满足M性质当且仅当它为相容FS-domain.     

H-代数偏序集

本文引入了H-代数偏序集的概念,讨论了它的一些基本性质.得到如下主要结果:(1)举例说明了代数偏序集未必是H-代数偏序集;(2)偏序集是H-代数偏序集当且仅当强紧元是它的强基;(3)偏序集是H-代数偏序集当且仅当它的局部Scott拓扑是强代数格.     

关于连续偏序集的一个刻画定理(英文)

本文给出了连续偏序集的一个刻画定理及相应的代数偏序集刻画定理.进一步得出连续偏序集P关于其Scott拓扑是局部紧的,而且OFilt(P)是一个domain.这推广了关于dcpo的对应结果.     

测度拓扑和连续偏序集的刻画

研究偏序集上的测度拓扑以及与其它内蕴拓扑间的关系,利用测度拓扑刻画了偏序集的连续性.构造了反例说明存在完全分配格,其上的测度拓扑不是连续格从而不是局部紧拓扑.     

伪超连续偏序集

引入了伪超连续偏序集和伪超代数偏序集的概念,讨论了它们的一些基本性质,给出了它们的若干刻画,特别是基于区间拓扑、某种特殊二元关系的刻画和内蕴式刻画.     

偏序集上一致连续性的等价刻画与性质

将一致小于关系移植到一般偏序集上,同时引入了上界小于关系,定义了偏序集的一致连续性和上界连续性.给出了一致连续偏序集的等价刻画,探讨了一致连续偏序集所具有的性质.主要结果有:(1)证明了偏序集上的一致连续性,上界连续性与s-超连续性均等价;(2)在交半格条件下,偏序集的一致连续性等价于它的每一主理想一致连续;(3)在并半格条件下,偏序集的一致连续性蕴含连续性,反之不成立;(4)一致完备的一致连续偏序集均是连续bc-dcpo,且每个主理想均为完全分配格;(5)在一致完备的条件下,一致连续性对主滤子,对闭区间,对Scott S-集以及对一致连续投射像均是可遗传的.文中也构造了若干实用的反例.     

可数S_2-拟连续偏序集北大核心CSCD

作为广义可数逼近偏序集与S2-拟连续偏序集的共同推广,引入了可数S2-拟连续偏序集的概念并讨论了它的一些性质.本文的主要结果:(1)可数S2-拟连续偏序集上的可数way below关系满足插入性质;(2)可数S2-拟连续偏序集关于其上的弱σ-Scott拓扑为局部紧致的可数sober空间;(3)偏序集P为可数S2-连续偏序集当且仅当P为可数S2-交连续的可数S2-拟连续偏序集.     

B-posets, FS-posets and Relevant Categories

In this paper the new concept of B-posets is introduced. Some properties of B-posets and FS-posets are examined. Main results are: (1) Posets obtained from B-posets (FS-posets) by eliminating a proper upper subset, adding two or more finitely many incomparable maximal elements, taking vertical sums w.r.t. a maximal element are also B-posets (FS-posets); (2) A poset is a(n) B-domain (FS-domain) iff it is a Lawson compact B-poset (FS-poset); (3) The directed completions of B-posets (FS-posets) are B-domains (FS-domains); (4) The category B-POS (FS-POS) of B-posets (FS-posets) and Scott continuous maps is cartesian closed and has the category B-DOM (FS-DOM) of B-domains (FS-domains) and Scott continuous maps as a full reflective subcategory.     

Metric projection onto finite-dimensional subspaces of C and L

In the space C(Q) of real functions that are continuous on the compact set Q, a finite-dimensional subspace P will have a uniformly continuous metric projection if and only if Q is a finite sum of compact sets Qi, and either P is on each Qi a one-dimensional Chebyshev space, or x(t)≡0 for any x belonging to P. The metric projection onto any finite-dimensional subspace of the space L[a, b] of real integrable functions is not uniformly continuous.     

Abstract ordered compact convex sets and algebras of the (sub)probabilistic powerdomain monad over ordered compact spaces

The majority of categories used in denotational semantics are topological in nature. One of these is the category of stably compact spaces and continuous maps. Previously, Eilenberg–Moore algebras were studied for the extended probabilistic powerdomain monad over the category of ordered compact spaces X and order-preserving continuous maps in the sense of Nachbin. Appropriate algebras were characterized as compact convex subsets of ordered locally convex topological vector spaces. In so doing, functional analytic tools were involved. The main accomplishments of this paper are as follows: the result mentioned is re-proved and is extended to the subprobabilistic case; topological methods are developed which defy an appeal to functional analysis; a more topological approach might be useful for the stably compact case; algebras of the (sub)probabilistic powerdomain monad inherit barycentric operations that satisfy the same equational laws as those in vector spaces. Also, it is shown that it is convenient first to embed these abstract convex sets in abstract cones, which are simpler to work with. Lastly, we state embedding theorems for abstract ordered locally compact cones and compact convex sets in ordered topological vector spaces.     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

连续自映射及其扭扩半流的不变测度

本文研究了紧致度量空间上连续自映射及连续半流的不变测度,并且证明了如下结论:(1)在拓扑等价的无不动点的连续半流的不变测度之间以及在连续自映射及其扭扩半流的不变测度之间存在一一对应;(2)作为(1)的应用,给出如下结论(见[2,定理2.1]):“环面上无不动点的连续流是唯一遍历的当且仅当它至多有一条周期轨”一个易接受的证明.     

Function-space compactifications of function spaces

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.     

Yosida frames

A Yosida frame is an algebraic frame in which every compact element is a meet of maximal elements. Yosida frames are used to abstractly characterize the frame of z-ideals of a ring of continuous functions C(X), when X is a compact Hausdorff space. An algebraic frame in which the meet of any two compact elements is compact is Yosida precisely when it is “finitely subfit”; that is, if and only if for each pair of compact elements a<b, there is a z (not necessarily compact) such that a∨z<1=b∨z. This is used to prove that if L is an algebraic frame in which the meet of any two compact elements is compact, and L has disjointification and dim(L)=1, then it is Yosida. It is shown that this result fails with almost any relaxation of the hypotheses. The paper closes with a number of examples, and a characterization of the Bézout domains in which the frame of semiprime ideals is Yosida frame.     

Order Continuity of Locally Compact Boolean Algebras

The main purpose of this paper is to exhibit the decisive role that order continuity plays in the structure of locally compact Boolean algebras as well as in that of atomic topological Boolean algebras. We prove that the following three conditions are equivalent for a topological Boolean algebra B: (1) B is compact; (2) B is locally compact, Boolean complete, order continuous; (3) B is Boolean complete, atomic and order continuous. Note that under the discrete topology any Boolean algebra is locally compact.     

Arzelà's Theorem and strong uniform convergence on bornologies

In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer-Levi conditions. The proof reveals the internal gear of these important convergences and sheds more light on the problem. We also study the main properties of the topology of strong uniform convergence of functions on bornologies, initiated in Beer and Levi (2009) [10].     

Ordinals and set-valued zero-selections for hyperspaces

A continuous zero-selection f for the Vietoris hyperspace F(X) of the nonempty closed subsets of a space X is a Vietoris continuous map f:F(X)→X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F(X) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection.