ABSTRACT. We show that convergence spaces with continuous maps and metric spaces with contractions, can be viewed as entities of the same kind. Both can be characterized by a "limit function" % which with each filter associates a map % from the underlying set to {ne extended positive real line. Continuous maps and contractions can both be (htacterized as limit function preserving maps.lhe properties common to both the convergence and metric case serve as a basis for tne definition of the category, CAP. We show that CAP is a quasitopos and that, apart from the categories CONV, of convergence spaces, and MET, of metric spaces, it also contains the category AP of approach spaces as nicely embedded subcategorles.
Abstract.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 {£} 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 oembedded 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 sé be a topological category. All subcategories 38 are assumed to be full and isomorphim-closed. A subcategory 38 of sé is epireflective in sé if it is closed with respect to the formation of products and subobjects in sé . 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 sé is contained in a smallest epireflective subcategory, its epireflective hull, which is denoted by R%. An object A of sé belongs to R% if and only if A is a subobject of a product of objects of <£ . A subcategory 33 of sé is called sé -simple if there exists a single object E of 38 such that 38 is the epireflective hull in sé of the class {E} , i.e., 38 = R{E} .Several examples of this situation are well known. If we take sé = 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.
PreliminariesWe assume familiarity with most of the usual notational conventions in the field and limit ourselves to recalling only the most important concepts and definitions required in this paper.Our basic category is FTS which stands for the topological category [l] of fuzzy topological spaces and continous maps as introduced in [3]. MAX stands for the full isomorphism closed subcategory of l?TS which was essentially introduced in [9]. In order to reformulate its definition we recall the notion of level topologies introduced in [5]. J i (X, A ) E lFTSl and a E I, then the a-level topology of A is defined as r.(d) := {p-I(]a, 11) I p E A } . Then by definition (X, A ) E (MAX( if and only if for any other fuzzy topology r on X with level topologies identical to thosa of d, i.e. such that &.(A) = c,(P) for all a E I,, it follows that I'c d.If no confusion can occur, the topological space (X, &.(A)) is often shortly denoted FNS stands for the full isomorphism closed subcategory of FTS coneisting of those fuzzy topological spaces which are generated by a fuzzy neighborhood system. This category waa introduced in [6], and the objects are also called fuzzy neighborhood spaces. We recall that a fuzzy neighborhood system on a set X consists of a collection of prefilters ( J~( z ) ) , , ,~where -denotes the so-called saturation operation defined by b:= (4 I v E E I,, 3 4, E S:(#l, -E 5 4}.The family ( Y , ) , ,~ in (FN3) is called an €-kernel for Y.
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