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

Abstract

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.