Semiuniform Convergence Spaces Research Articles

A one to one correspondence between connected varieties and disconnected varieties of ⊤-semiuniform convergence spaces

In this paper, we generalize Preuss' E-connectedness of semiuniform convergence spaces from classical to lattice-valued cases with the underlying lattice being a unital commutative quantale. We propose the concept of E-connected ⊤-semiuniform convergence spaces for a subclass E of ⊤-semiuniform convergence spaces and that of connected varieties in the sense that a connected variety precisely is the class consisting of all E-connected spaces for some subclass E. Then a one to one correspondence between connected varieties and disconnected varieties is established. And further, we obtain an internal characterization of connected varieties and that of disconnected varieties, respectively.

Monoidal closedness of the category of $\top$-semiuniform convergence spaces

Lattice-valued semiuniform convergence structures are important mathematical structures in the theory of lattice-valued topology. Choosing a complete residuated lattice $L$ as the lattice background, we introduce a new type of lattice-valued filters using the tensor and implication operations on $L$, which is called $\top$-filters. By means of $\top$-filters, we propose the concept of $\top$-semiuniform convergence structures as a new lattice-valued counterpart of semiuniform convergence structures. Different from the usual discussions on lattice-valued semiuniform convergence structures, we show that the category of $\top$-semiuniform convergence spaces is a topological and monoidal closed category when $L$ is a complete residuated lattice without any other requirements.

A COMMON IMPROVEMENT OF FUZZY TOPOLOGICAL SPACES AND FUZZY (QUASI) UNIFORM SPACES

Using fuzzy filters in the sense of P. Eklund and W. Gahler [2], it turns out that fuzzy preuniform convergence spaces introduced in [11] form a strong topological universe in which fuzzy topological spaces as well as fuzzy (quasi) uniform spaces can be studied. Thus, better tools such as the existence of natural function spaces, the existence of one-point extensions (and consequently, the hereditariness of quotient maps), and the productivity of quotient maps are available. 0 Introduction. In 1968 fuzzy topological spaces have been introduced by C.L. Chang [1]. Together with the fuzzy continuous maps between them they form a topological construct provided that in the definition of a fuzzy topological space X the requirement is incorporated that all fuzzy subsets of X given by constant maps are fuzzy open. This has been pointed out by R. Lowen [8]. Concerning fuzzy filters, in this paper a definition due to P. Eklund and W. Gahler [2] is used which fuzzificates additionally the membership of filter elements. This leads to an alternative definition of fuzzy uniform spaces introduced by W. Gahler et al. [6] in 1998. Omitting a certain symmetry condition in this definition one obtains fuzzy quasiuniform spaces analogously to the non-fuzzy case. In 2005 the author [10] studied preuniform convergence spaces which form a strong topological universe, i.e. a topological construct which is 1 cartesian closed (i.e. natural function spaces exist), 2 extensional (i.e. one-point extensions exist), and in which 3 (arbitrary) products of quotients are quotients. Furthermore, they are suitable for generalizing topological spaces as well as quasiuniform spaces. This is very remarkable since neither topological spaces nor (quasi) uniform spaces fulfill the above mentioned convenient properties 1, 2, and 3 with the following exception: 3 is true for uniform spaces, and it is unknown whether 3 is true for quasiuniform spaces (cf. [10]). The aim of this paper is to realize that the fuzzyfication of preuniform convergence spaces which has been started in [11] leads to a strong topological universe too, and thus improves fuzzy topological spaces and fuzzy (quasi) uniform spaces. Since in non-symmetric convenient topology (cf. [10]) mainly preuniform convergence spaces are investigated we are now in the position to have a suitable framework for non-symmetric fuzzy convenient topology. Finally, adding a certain symmetry condition to the definition of a fuzzy preuniform convergence space we obtain a fuzzy semiuniform convergence space, whose non-fuzzy analogue is mainly studied in convenient topology (cf. [9]). Since the construct FSUConv of fuzzy semiuniform convergence spaces is closely related to the construct FPUConv of fuzzy preuniform convergence spaces, it results that is a strong topological universe too. Therefore, the foundations of fuzzy convenient 2003 Mathematics Subject Classification. 54A40, 18A40, 18D15.

Stratified (L; M)-semiuniform convergence tower spaces

The notion of stratified (L, M)-semiuniform convergence tower spaces is introduced, which extends the notions ofprobabilistic semiuniform convergence spaces and lattice-valued semiuniform convergence spaces. The resulting categoryis shown to be a strong topological universe. Besides, the relations between our category and that of stratified (L, M)-filter tower spaces are studied.

Closure operators in semiuniform convergence spaces

In this paper, the characterization of closed and strongly closed subobjects of an object in category of semiuniform convergence spaces is given and it is shown that they induce a notion of closure which enjoy the basic properties like idempotency,(weak) hereditariness, and productivity in the category of semiuniform convergence spaces. Furthermore, T1 semiuniform convergence spaces with respect to these two new closure operators are characterized.

Lattice-valued semiuniform convergence spaces

In this paper, two kinds of lattice-valued semiuniform convergence spaces are proposed, namely stratified L-semiuniform convergence spaces and stratified L-ordered semiuniform convergence spaces respectively. It is shown that (i) the category of stratified L-semiuniform convergence spaces is topological; (ii) the category of stratified L-ordered semiuniform convergence spaces is a bireflective full subcategory of the category of stratified L-semiuniform convergence spaces, and hence it is topological; (iii) both the category of stratified L-semiuniform convergence spaces and that of stratified L-ordered semiuniform convergence spaces are Cartesian-closed; (iv) the category of stratified L-semiuniform convergence spaces is extensional; (v) both the category of stratified L-semiuniform convergence spaces and that of stratified L-ordered semiuniform convergence spaces are closed under the formation of products of quotient mappings. In case that L is the two-point chain, both coincide with the category of semiuniform convergence spaces in the classical case.

A better framework for first countable spaces

<p>In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.</p>

Completion of Probabilistic Uniform Limit Spaces

In this article completions of special probabilistic semiuniform convergence spaces are considered. It turns out that every probabilitic Cauchy space under a given t-norm T (triangular norm) has a completion which, in the special case of probabilistic Cauchy spaces with reference to T=min, coincides with the KentRichardson completion for probabilistic Cauchy spaces. Moreover, a completion of probabilistic uniform limit spaces wrt T=min is given which in case of constant probabilistic uniform limit spaces coincides with the Wyler completion.

A Generalization of Probabilistic Uniform Spaces

We develop a theory for probabilistic semiuniform convergence spaces. Probabilistic semiuniform convergence spaces generalize probabilistic uniform spaces in the sense of Florescu and probabilistic convergence spaces in the sense of Kent and Richardson. This theory includes a new branch in topology, namely, Convenient Topology, introduced by Preus. Thus, it includes semiuniform convergence spaces and uniform spaces, filter and Cauchy spaces and (symmetric) limit spaces and, therefore, (symmetric) topological spaces. The theory of probabilistic semiuniform convergence spaces reveals categories which are strong topological universes or have other convenient properties.

The topological universe of locally precompact semiuniform convergence spaces

The importance of local precompactness has become apparent in theorems of the Ascoli type proved by Bentley and Herrlich (1985) in the realm of filter spaces as well as by Wyler (1984) in the realm of uniform limit spaces. Here local precompactness is studied in the highly convenient setting of sem iuniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces), filter spaces and uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a topological universe. Furthermore, locally precompact semiuniform convergence spaces are exactly the precompactly generated semiuniform convergence spaces, and the relationships to locally compact, compact and precompact semiuniform convergence spaces respectively, studied before by Preuß (1995), are clarified.

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.1