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The fact that approach spaces generalize at the same time topological and metric spaces implies that we can consider at the same time both a notion of completion and a notion of compactification, especially for uniform approach spaces. Further we also consider a completion for uniform gauge spaces. All these extensions turn out to be epireflections in their respective settings and hence also give rise to natural extensions of morphisms.

Both topological and metric spaces can be viewed as special types of approach spaces. More precisely, both the categories of topological spaces and continuous maps, and of (quasi)-metric spaces and nonexpansive maps, can be embedded as full and isomorphism-closed subcategories of the category of approach spaces, the former as a stable (i.e. simultaneously concretely reflective and concretely coreflective) subcategory and the latter as a concretely coreflective subcategory.

Here we study approach structures (or uniform gauge structures) on hyperspaces of closed sets of metric spaces. In the first section we study a natural quantification of the Wijsman topology, in the second section we study the proximal topologies and in the last example, we study a quantified version of the Vietoris structure in the more general setup of closed sets in an arbitrary approach space.

In Chaps. 7– 11 we study applications of approach theory to various areas of mathematics. Here we highlight two areas in topology, namely function spaces and the Cech-Stone compactification. In the first section we treat function spaces, starting from arbitrary approach spaces and uniform gauge spaces. The results are applicable to any situation where for instance a choice has been made of a particular metric for a topology or a uniformity. In the second section we study the approach version of the Cech-Stone compactification of an Atsuji space, and in particular of the natural numbers, in more detail.

Here we propose an intrinsic solution for the problem of quantifiability in domain theory. We construct an approach structure which quantifies the Scott topology regardless of cardinality conditions on bases. We show that every domain is quantifiable in this sense. We get weightability for free and in the case of an algebraic domain satisfying the Lawson condition, a quantifying approach space can be obtained with a weight satisfying the so-called kernel condition. With respect to contractions, we study fixed point theorems. For monotone as well as for non-monotone maps we establish new fixed point theorems and we recover several existing ones as special cases.

Here we start in the classical setting of normed spaces and see how the well-known weak and weak* topologies can be quantified by canonical approach structures and how using index analysis, this allows to obtain quantified results of which several classical results are simple corollaries. In a second part we see that the construction of the weak and weak* approach structures on normed spaces fits into the wider picture of what we call approach vector spaces and locally convex approach spaces.

By means of a technique based on probabilistic considerations, we give a number of new and rediscover some old connectives predominantly used in fuzzy set theory. We construct these connectives from the connectives of classical logic by means of what one might call a simple “probabilistic black box.”

(1991). COMMENTS ON ZELENÝ'S PAPER CONCERNING COGNITIVE EQUILIBRIA. International Journal of General Systems: Vol. 19, No. 4, pp. 387-393.

In this paper we prove a cancellation law in the category (1) of partially ordered sets and (2) of To spaces, the respective conditions on the common factor being that it have (1) an irreducible bottom element and (2) an irreducible bottom element in its specialization order.

Before embarking on our journey let us point out what we will not be doing, thereby stifling unjustified “great expectations” in the bud.

As the name suggests, fuzzy sets are related to sets. We shall therefore first of all give a short overview of the basics of “elementary set theory”. It is not our purpose to introduce these concepts in a highly rigorous way, rather we like to present this material in a leisurely fashion, adapted to our purposes.

Fuzzy real numbers form an important concept in all applications of fuzzy set theory. There are several different definitions of what a fuzzy real number could be, to be found in the literature. We shall give an account of the most common and most important types, together with the operations defined on them.

To describe the current use, effectiveness and safety of Bier's block within Australian emergency departments.A review of clinical practice based on an opinion survey of directors of emergency departments.Directors of 120 private and public hospital emergency departments.A response rate of 97% was achieved. Bier's block was used in 67% of Australian emergency departments. Emergency department medical staff performed the procedure in 83% of these hospitals. It was estimated that 7500 emergency department Bier's blocks were performed each year. Ninety-six per cent of respondents found Bier's block to produce adequate regional anaesthesia. Reported complications or adverse reactions were extremely rare. A wide variation in procedural approach was evident.Bier's block is a popular, reliable and relatively safe anaesthetic technique for use in emergency departments. Increased surveillance of local anaesthetic morbidity associated with the technique, as well as a standardisation of the procedure in the emergency department setting, are required.

In this paper we identify for a fuzzy topological space (X, Δ) a collection V(X) of new points, some or all of which can be appended to X in order to produce various compactifications of X. Specifically, we consider compactification and ultra-compactification of (X, Δ). V(X) has its roots in Shanin’s generalization of Wallman’s construction as given in [14]. Our construction is analogous to (Shanin’s generalization of) Wallman’s compactification for topological spaces in that we explicitly describe the closed sets and convergence of prefilters in the compactification; in fact, it coincides with Wallman’s compactification if (X, Δ) is topologically generated. Thus our compactifications are good extensions of Wallman’s compactification to the fuzzy setting. Our construction coincides with (and extends) the corresponding compactifications given in [1] for fuzzy neighborhood spaces. However, the constructions given here apply to all fuzzy topological spaces; no separation properties are required nor is (X, Δ) restricted in any way. Thus our construction has advantages over previous fuzzy compactifications in e.g. [1], [2], [4], and [12], which apply to a smaller class of fuzzy spaces (sometimes only to topologically generated spaces).

E.G. Manes gave a characterization of the category of topological spaces as a suitable category of sets with structure having an appropriate “Sierpinski object”. By identifying an appropriate Sierpinski object, A.K. Srivastava and R. Srivastava obtained an analogous characterization of the category of fuzzy topological spaces. In the present paper, we extend these ideas further and give similar characterizations of the categories of fuzzy neighbourhood spaces and topological fuzzy topological spaces by identifying appropriate Sierpinski objects in these categories.

This paper presents a framework in which basic concepts of approximation theory arise as canonically as convergence in topological spaces.

The category TOP of topological spaces is known to be Convsimple in the sense that there exists, a single object E in Top such that Top is the epireflective hull of {E} in the category Conv of convergence spaces. Prtop, the category of pretopological spaces is also Conv-simple. We show that on the contrary the category Pstop of pseudo topological spaces and Conv itself are not Conv-simple. More specifically every epireflective subcategory of Conv which contains all Hausdorff c-embedded locally compact spaces is not Conv-simple. For references on topological categories, reflective subcategories and reflective hulls we refer to [6, 8, 9, 10]. Let v be a topological category. All subcategories ~W are assumed to be full and isomorphim-closed. A subcategory W of v is epireflective in v if it is closed with respect to the formation of products and subobjects in v . In this context " Y is a subobject of X" means that there exists an embedding from X to Y. This notion coincides with the categorical notion of extremal subobject. Every subcategory ' of v is contained in a smallest epireflective subcategory, its epireflective hull, which is denoted by R?. An object A of v belongs to R? if and only if A is a subobject of a product of objects of F'. A subcategory W of v is called X -simple if there exists a single object E of w such that q is the epireflective hull in v of the class {E}, i.e., 97 = R{E} . Several examples of this situation are well known. If we take v = Conv then its bireflective subcategory Top is Conv-simple. Several subcategories of TOP are Conv-simple too, see for instance [4, 5, 6, 7, 9, 14]. Simplicity remains true if TOP is enlarged to the bireflective subcategory Prtop. This can be derived from results in [1]. In this paper we show that simplicity however does not extend to the larger bireflective subcategories Pstop or to Conv itself. Received by the editors December 4, 1987 and, in revised form, April 11, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 18B99, 54B30, 54A25. This paper was presented at the conference "Convergence structures in topology and analysis" at Mathematiches Forschungs Institute Oberwolfoch, December 1987. ? 1989 American Mathematical Society 0002-9939/89 $1 00 + $.25 per page

Let X X be a uniform topological space, then on the family I X {I^X} (resp. Φ ( X ) \Phi (X) ) of all nonzero functions (resp. nonzero uppersemicontinuous functions) from X X to the unit interval I I , a fuzzy uniform topology is constructed such that 2 X {2^X} (resp. F ( X ) \mathcal {F}(X) ), the family of all nonvoid (resp. nonvoid closed) subsets of X X equipped with the Hausdorff-Bourbaki structure is isomorphically injected in I X {I^X} (resp. Φ ( X ) \Phi (X) ). The main result of this paper is a complete description of convergence in I X {I^X} , by means of a notion of degree of incidence of members of I X {I^X} . Immediate consequences are that first it can be shown that this notion of convergence refines some particular useful notions of convergence of fuzzy sets used in applications, and that second it follows from its construction and properties that for each ordinary uniform topological space X X there exists a natural nontopological fuzzy uniform topology on I X {I^X} .

During the Vth Seminar on Fuzzy Set Theory (Linz, Austria, September 1983) U. Cerruti and U. Hohle stated the following open problem from their work on fuzzy equalities. “16 (X,E) is a set with [0,1]-valued fuzzy equality E, and (P(X),E) is the set of fuzzy points with the induced fuzzy equality E, what is the precise relationship between X and P(X) and between E and E?” In particular they were unable to show that P(X) in general is “larger” than X and wondered if each fuzzy point ρ was of the type ρ(z) = E(x, z) for some x e X. During the seminar we solved this question in the negative by the example following Corollary 2.3. based on [3] and conjectured that it could be generalized to arbitrary pseudometric spaces. In this paper we prove this conjecture. We show that for a large class of fuzzy equalities the relationship between X and P(X) is as between a pseudometric space and its completion and between E and [Etilde] is as between a uniformly continuous map and its continuous extension. A first proof i...