Compact Hausdorff fuzzy topological spaces are topological

Many examples of compact fuzzy topological spaces which are highly non topological are known [5, 6]. Equally many examples of Hausdorff fuzzy topological spaces which are highly non topological can be given. In this paper we show that the two properties - compact and Hausdorff - combined however necessarily imply that the fuzzy topological space is topological. This at once solves some open questions with regard to the compactification of fuzzy topological spaces [8]. It also emphasizes once more the particular role played by compact Hausdorff topological spaces not only in the category of topological spaces but even in the category of fuzzy topological spaces.     

Expanding topological space,study and applications

In this paper, we introduce the notion of expanding topological space. We define the topological expansion of a topological space via local multi-homeomorphism over coproduct topology, and we prove that the coproduct family associated to any fractal family of topological spaces is expanding. In particular, we prove that the more a topological space expands, the finer the topology of its indexed states is. Using multi-homeomorphisms over associated coproduct topological spaces, we define a locally expandable topological space and we prove that a locally expandable topological space has a topological expansion. Specifically, we prove that the fractal manifold is locally expandable and has a topological expansion.     

Algebraic objects generated by topological structure

The survey examines the properties of the Tikhonov space X that depend on the topological properties of the function space C(X) with pointwise topology and of the free topological group F(X). It covers the part of the theory of topological groups dealing with topological invariants and topological classification of topological groups. Continuous images of topological groups are considered in the context of studying the actions of topological groups on spaces.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 25, pp. 141–198, 1987.     

拓扑系统的紧性和分离性

考察拓扑系统的两种紧性——空间式紧和locale式紧,给出紧性的若干刻画,讨论了两种紧性的相互关系,证明了拓扑系统的两种紧性都是拓扑空间紧性的良好推广,说明了紧拓扑系统的闭子拓扑系统、有限和系统以及积系统仍是紧拓扑系统。最后在拓扑系统中考察了紧性加强分离性的问题,得到了紧,(强)T2拓扑系统为(强)T3,(强)T4拓扑系统等结论,并用理想收敛刻画了拓扑系统的强T2分离性。     

Coordinate rings of topological Klingenberg planes I: The affine perspective

Although the coordinate ternary field of a topological affine plane is topological, the converse does not hold. However, an affine plane is topological precisely when its coordinate biternary fields are topological. We extend this result to topological biternary rings and their topological affine Klingenberg planes. Then we examine the locally compact situation. Finally, following the ideas of Knarr and Weigand, we show that in certain circumstances, the continuity of the ternary operators is sufficient to ensure that the biternary ring is topological. This facilitates the construction of locally compact, locally connected affine Klingenberg planes.Dedicated to Professor Dr. Helmut Salzmann on his 65th birthday     

关于模糊拓扑环的直积

给出模糊拓扑环族直积的定义,研究它的性质,及任意模糊拓扑环族直积、与其模糊拓扑子环,与其模糊拓扑剩余环之间的同构问题.并对(QU)型模糊拓扑环关于上述问题进行研究.     

Locale的弱拓扑表达

本文引入了弱拓扑空间的概念,证明了locale范畴与弱拓扑空间范畴的关系类似于拓扑空间范畴与locale范畴的关系。locale范畴严格包含于弱拓扑空间范畴并且与Sober的弱拓扑空间范畴等价。     

On topological Clifford semigroups embeddable into products of cones over topological groups

In this paper we give characterizations of topological Clifford semigroups which are embeddable into Tychonoff products of topological semilattices and cones over topological groups. Also we characterize topological Clifford semigroups which embed into compact topological Clifford semigroups.     

半拓扑系统

本文旨在建立Ｓ系统理论，它具有相当的广泛性，以及应用背景性，同时又以拓扑空间、模糊拓扑空间、拓扑分子格以及拓扑系统为特例．     

Embedding topological semigroups in topological groups

In [6] Rothman investigated the problem of embedding a topological semigroup in a topological group. He defined a concept calledProperty F and showed that Property F is a necessary and sufficient condition for embedding a commutative, cancellative topological semigroup in its group of quotients as an open subset. This paper announces a generalization of Rothman’s result by definingProperty E and stating that a completely regular topological semigroup S can be embedded in a topological group by a topological isomorphism if and only if S can be embedded (algebraically) in a group and S has Property E. Property E is defined by first constructing a free topological semigroup (Theorem 1.1). This construction resembles the one in [4] for a free topological group. Full details, examples, and other embedding results will appear elsewhere. Some of the results in this paper were contained in the author’s doctoral dissertation written at Rutgers University under Professor Louis F. McAuley.     

关于几乎拓扑群的注记

In this paper, we mainly discuss some generalized metric properties and the cardinal invariants of almost topological groups. We give a characterization for an almost topological group to be a topological group and show that:(1) Each almost topological group that is of countable π-character is submetrizable;(2) Each left λ-narrow almost topological group isλ-narrow;(3) Each separable almost topological group is ω-narrow. Some questions are posed.     

Coordinate rings of topological Klingenberg planes II: the algebraic foundation for a projective theory

Ternary fields are the coordinate rings of affine and projective planes; however, the planes constructed over topological ternary fields are not necessarily topological. Surprisingly, the explanation of this phenomenon becomes evident in the more general theory of topological Klingenberg planes as we exhibited in [3] for the affine case. However, in the projective setting, we have a more formidable task. We must develop a new coordinate ring that admits a topological structure suitable for coordinatizing topological PK-planes. We accomplish this in two stages. In this paper, we revisit the standard coordinate rings [1, 11], discuss and resolve their deficiencies by developing a new coordinate ring as a unique extension of these refined standard rings. In a subsequent paper [4], we show that this new ring can be suitably topologized to coordinatize a topological PK-plane. This last result can then be used to explain why topological ternary fields do not necessarily coordinatize topological projective planes. Received 17 February 2000; revised 10 June 2000.     

灰拓扑群

给出了灰拓扑群的定义及有关定理,在此基础上讨论灰拓扑群与Fuzzy拓扑群、一般拓各之关系,并且研究了灰拓扑群的子群。     

On Chains in <Emphasis Type="BoldItalic">H</Emphasis>-Closed Topological Pospaces

We study chains in an H-closed topological partially ordered space. We give sufficient conditions for a maximal chain L in an H-closed topological partially ordered space (H-closed topological semilattice) under which L contains a maximal (minimal) element. We also give sufficient conditions for a linearly ordered topological partially ordered space to be H-closed. We prove that a linearly ordered H-closed topological semilattice is an H-closed topological pospace and show that in general, this is not true. We construct an example of an H-closed topological pospace with a non-H-closed maximal chain and give sufficient conditions under which a maximal chain of an H-closed topological pospace is an H-closed topological pospace.     

Universal topological algebra needs closed topological categories

It is shown that a development of universal topological algebra, based in the obvious way on the category of topological spaces, leads in general to a pathological situation. The pathology disappears when the base category is changed to a cartesian closed topological category or to a topological category endowed with a compatible closed symmetric monoidal structure, provided that in the latter case, the algebraic operations are expressed in terms of monoidal powers rather than the usual cartesian powers. With such base categories, universal topological algebra becomes virtually as well-behaved as ordinary (setbased) universal algebra.     

赋层拓扑分子格及其紧性

拓扑分子格本身没有层次结构，为其构造某种层次成为很重要的问题。本文的构思是赋于分子格上的拓扑一种层次，从而使拓扑分子格具有丰富的层次性，其次讨论了保层次序同态与赋层积拓扑分子格，在此基础上建立了较理想的拓扑分子格的紧性。     

A Homological Version of the Implicit Function Theorem

Our objective in this paper is to prove an Implicit Function Theorem for general topological spaces. As a consequence, we show that, under certain conditions, the set of the invertible elements of a topological monoid X is an open topological group in X and we use the classical topological group theory to conclude that this set is a Lie group.     

拓扑序空间中的广义向量值均衡问题

利用一个KKM型定理,在拓扑序空间中,建立了关于广义向量值均衡问题解的某些新的存在性定理.     

开广义序同态的刻划及拓扑分子格的特征与权

王国俊先生于〔１〕中借助闭元成功地引入了开广义序同态的概念。本文给出了开广义序同态的一个刻划并对其是否增加拓扑分子格的权与特征等问题进行了研究，得到了积拓扑分子格的特征与权与因子拓扑分子格的特征与权间的关系，从而部分回答了ＬＦ拓扑中关于特征的两个公开问题。