Strongly continuous posets and the local Scott topology

In this paper, the concept of strongly continuous posets (SC-posets, for short) is introduced. A new intrinsic topology—the local Scott topology is defined and used to characterize SC-posets and weak monotone convergence spaces. Four notions of continuity on posets are compared in detail and some subtle counterexamples are constructed. Main results are: (1) A poset is an SC-poset iff its local Scott topology is equal to its Scott topology and is completely distributive iff it is a continuous precup; (2) For precups, PI-continuity, LC-continuity, SC-continuity and the usual continuity are equal, whereas they are mutually different for general posets; (3) A T₀-space is an SC-poset equipped with the Scott topology iff the space is a weak monotone convergence space with a completely distributive topology contained in the local Scott topology of the specialization order.     

偏序集上的测度拓扑与稠拓扑

In this paper,the concepts of the essential topology and the density topology of dcpos are generalized to the setting of general posets.Basic properties of the essential topology and relations with other intrinsic topologies are explored.Comparisons between the density topology and the measurement topology are made.Via the essential topology,the density topology and the measurement topology,we obtain properties and characterizations of bases of continuous posets.We also provide some new conditions for a continuous poset to be an algebraic poset.     

Notes on the products of the lower topology and Lawson topology on posets

In this paper, we investigate the relation between the lower topology respectively the Lawson topology on a product of posets and their corresponding topological product. We show that (1) if S and T are nonsingleton posets, then Ω(S×T)=Ω(S)×Ω(T) iff both S and T are finitely generated upper sets; (2) if S and T are nontrivial posets with σ(S) or σ(T) being continuous, then Λ(S×T)=Λ(S)×Λ(T) iff S and T satisfy property K, where for a poset L, Ω(L) means the lower topological space, Λ(L) means the Lawson topological space, and L is said to satisfy property K if for any x∈L, there exist a Scott open U and a finite F⊆L with x∈U⊆↑F.     

Flat topology and its dual aspects

In this article, a new and natural topology on the prime spectrum is introduced, which behaves completely as the dual of the Zariski topology. It is called the flat topology. The basic and also some sophisticated properties of the flat topology are proved. Specially, various algebraic characterizations for the noetherianess of the flat topology are given. Using the flat topology, then some facts on the structure of the prime ideals of a ring come to light which are not in the access of the Zariski topology.     

Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

Posets admitting a unique order-compatible topology

A characterization is given for those posets (X, ?) such that X admits exactly one topology inducing the given partial order ?. As a corollary, a poset is finite if and only if it is finite-dimensional and admits a unique compatible topology. Related applications and examples are also developed.     

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.     

On fuzzy compact-open topology

The concept of fuzzy compact-open topology is introduced and some characterizations of this topology are discussed.     

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.     

5元素集合上T_0拓扑总数的计算

利用有限偏序集上的几个重要结果并借助于拓扑空间对应的特殊化序与拓扑之间的关系计算得出5元素集合上T0拓扑总数为4231,拓扑总数为6942.     

Topologies on products of partially ordered sets III: Order convergence and order topology

This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are:

1. Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5).
2. For any product of posets, the projections are open and continuous with respect to the order topologies (2.1).
3. A productL of chainsL _i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology onL agrees with the product topology (2.7).
4. If (L _i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8).
5. LetP ₁ be a poset with topological order convergence and locally compact order topology. Then for any posetP ₂, the order topology ofP ₁?P ₂ coincides with the product topology (2.10).
6. A latticeL which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology ofL?L is the product topology (2.15).

Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.     

Fuzzifying拓扑中的Pre-网收敛

运用连续值逻辑语义的方法研究fuzzifying拓扑空间,从Pre-开集出发引入了Pre-导集的概念,并且给出了它的一些性质,进一步探讨了Pre-网收敛理论.这些研究有助于丰富和发展fuzzifying拓扑学的基本理论.     

Cartesian Closed Categories of F Z-domains

A subset systemZassigns to each partially ordered setPa certain collectionZ(P) of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Z, the concepts of F Z-way-below relation and F Z-domain are introduced. The well-known Scott topology is naturally generalized to the Z-level and the resulting topology is calledF Z-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Z-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS ofF Sdomains to the Z-level. Corresponding to them, it is proved that, for a suitable subset system Z, the categories FZCPO ofZ-complete posets,FSFZ of finitely separated FZ-domains andBFFZ of bifiniteF Z-domains are all cartesian closed. Some examples of these categories are given.     

Weak topology and infinite matrix games

This paper approaches infinite matrix games through the weak topology on the players' sets of strategies. A new class of semi-infinite and infinite matrix games is defined, and it is proved that these games always have a value and optimal strategies for each player. Using these games it is proved that some other important classes of infinite matrix game also have values. Received April 1996/Revised version June 1997/Final version September 1997     

Topologies on products of partially ordered sets II: Ideal topologies

Within the theory of ideals in partially ordered sets, several difficulties set in which do not occur in the special case of lattices (or bidirected posets). For example, a finite product of ideals in the factor posets need not be an ideal in the product poset. The notion ofstrict ideals is introduced in order to remedy some deficiencies occurring in the general case of an arbitrary product of posets. Besides other results, we show the following main theorem: The ideal topology (cf. [2]) of a product of non-trivial posets coincides with the product topology if and only if the number of factors is finite (4.19.). Presented by L. Fuchs     

Group topologies coarser than the Isbell topology

The Isbell, compact-open and point-open topologies on the set C(X,R) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α(X) of compact families of open subsets of a topological space X. Those α(X) for which addition is jointly continuous at the zero function in Cα(X,R) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α(X) for which Cα(X,R) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, that Cα(X,R) can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology.     

Interval topology on effect algebras

In this paper, we study the interval topology on effect algebras, and prove that effect algebra operation on Hilbert space effect algebra E(H) is not jointly continuous under the interval topology.     

ech-complete spaces and the upper topology

Let X be a topological space and let be the set of all compact subsets of X. The purpose of this note is to prove the following: if X is regular and q-space, then X is Lindelöf and ech-complete if and only if there exists a continuous map f from a Lindelöf and ech-complete space Y to the space endowed with the upper topology, such that f(Y) is cofinal in . This result extends the following result of Saint Raymond and Christensen: if X is separable metrizable, then X is a Polish space if and only if the space endowed with the Vietoris topology is the continuous image of a Polish space.     

Riesz operators and the complete norm topology

We show that a non-injective Riesz operator on an infinite-dimensional Banach space X does not determine the complete norm topology of X. We also show that an injective operator with trivial generalized range determines the complete norm topology of X. Finally this result is used to settle the crucial role of the non-injectivity condition in our first result.