CHARACTERIZATION OF CONVERGENCE IN FUZZY TOPOLOGICAL SPACES

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E. LOWEN AND R. LOWEN which is paramount and which most clearly demonstrates the aim of fuzzy topology. See e.g. fuzzy topologies on hyperspaces of fuzzy sets [12], [13], on metric spaces [17] and on spaces of probability measures [15], [16]. In each of these cases the notion of convergence in the fuzzy topology permits to "measure" a "degree" with which a filter converges to a point, and in each case maximal degree of convergence is equivalent to classical topological convergence in some associated topology. In the examples jus mentioned these are respectively the Hausdorff-Bourbaki hyperspace topology on closed sets [18], the metric topology, and the topology of weak convergence [i].
In the context of fuzzy topologies it is therefore equally important to characterize fuzzy topological convergence internally. In this paper we solve this question and give a set of 6 axioms which turn out to be necessary and sufficient for a fuzzy convergence to be fuzzy topological.
With regard to these axioms a number of comments are in order.
First, it turned out that the diagonal condition of Kowalsky cannot be translated in a straightforward manner. The classical condition of a filter being convergent to a point has no meaning and its substitute i.e. the information of the degree with which a filter converges to a point can only be handled analytically and thus has been incorporated as such in the diagonal condition. This "fuzzy" diagonal condition will play a key role in the characterization of convergence in fuzzy topological spaces.
Second, the fundamental classical condition concerning the convergence of comparable filters has to be replaced by two separate axioms.
The first axiom is analogous to the classical one with the exception that only filters which in a certain sense are "horizontally" comparable may be considered. Due to the fact however that prime filters are not necessarily maximal we also have a type of "vertical" comparability for filters and we need a second axiom to deal with those.
Finally, yet another axiom concerning the "overall" degree of convergence of a filter is required which is purely "fuzzy" in the sense that it has no classical meaning or counterpart.
At the end of our paper we restrict our 6 axioms to prime filters and show that these "prime versions" too already fully characterize fuzzy convergence. These prime versions are important because the convergence theory in fuzzy topological spaces is founded in large part on the use of prime filters.

PRELIMINARIES.
We recall that I :-[0,i] and that if X is a set I X denotes the set of all functions X I, i.e. all fuzzy sets on X, equipped with the usual lattice structure. As X s,ch I is a complete and completely distributive lattice. If X is a set and A C X then I A denotes the characteristic function of A and if A then we write i for 1 x {x}" In order to discern between filters on X and a type of "fuzzy filters" we shall, for the latter, simply use the term introduced by Bourbaki [3] in the framework of general lattices.
A prefilter (resp. prime prefilter) (on X) is a prefilter (resp. prime prefilter) in X X the lattice If u then we denote by the principal prefilter The set P(X) of all prefilters on X is ordered by inclusion and fulfils analogous order properties as the set F(X) of all filters on X. We use the same well-known notations and terminology for both.    As an immediate consequence remark that if 6 I X then for each x 6 X we can thus find { e P (6) such that (x) lim (x). m Finally, we introduce also the following concept which shall be of crucial importance in our considerations, since it is precisely the tool which permits to generalize Kowalsky's diagonal condition.
If (' (Y))y6X is a family of prefilters indexed by X -called a selection of prefiltersthen we define @ as the map . and this for all y X. Consequently D e and a fortiori p.
for any From the first assertion we then already obtain that lim -3(((y))y6X,l(t)) lira To conclude the proof of the theorem we shall now show that the map is idempotent.
Let 6 I X and 6 Pm() be fixed. Now remark that the proof of Proposition 2.12 only uses the fact that closure and limit in a fuzzy topological space fulfil (4.1), (4.2) and (4.5). Since we have already shown that indeed fulfils (4.1), (4.2) and (4.5), by means of a perfectly analogous proof, we can now ascertain that for each y e X there exists (y) 6 Pro(6) such that lira (y)(y) (y).

PROOF.
This goes the same as the proof of (FI) in

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