Separation in Ditopological Texture Spaces
by
Riza Ertürk
Mersin University
Coauthors: Lawrence M. Brown (Hacettepe University)

This study extends previous work [4] on the relationship between fuzzy topology and bitopological spaces. There it was shown that although a bitopology on the set S of fuzzy points of a fuzzy topological space can represent the fuzzy topology more faithfully than a topology on S, the match is still not exact. On the other hand in [2] it was shown that by working in terms of a suitable subfamily S of the powerset P(S), rather than in terms of P(S) itself, it is possible to represent the fuzzy sets precisely in set theoretic terms, and then by working in terms of the notion of ditopology rather than bitopology [3], we can get a perfect match.

These considerations have given rise to the concept of a texturing S of a set S, which is a point separating, complete, completely distributive lattice S of subsets of S with respect to inclusion, which contains S, \emptyset, and for which meet /\ coincides with intersection \cap and finite joins \/ coincide with unions \cup . Textures (S, S) corresponding to lattices of (L-) fuzzy sets have in addition the properties of being simple - every molecule (join irreducible element) of S has the form P_s, s in S, where P_s is the smallest element of S containing s - and of having a complementation, which a mapping \sigma:S --> S reversing inclusion and satisfying \sigma(\sigma(A))=A for all A in S. It has also given us the concept of ditopology. Just as a bitopology consists of two families of open sets, a ditopology consists of a family \tau of open sets containing S, \emptyset and closed under finite meets and arbitrary joins, and a family \kappa of closed sets containing S, \emptyset and closed under finite joins and arbitrary meets. In case S, S) has a complementation \sigma and \kappa = \sigma(\tau) the ditopology (\tau, \kappa) is said to be complemented.

In terms of the concepts defined above a fuzzy topological space corresponds to a simple complemented ditopological texture space (S, S, \sigma, \tau, \kappa). In this talk we consider separation properties for general ditopological texture spaces, and relate these to fuzzy separation axioms for the special case mentioned above. We find that most separation properties may be given two dual forms, such as regular and coregular, with some, like normality, being self dual. We show that for a complemented ditopology these dual properties coincide and that for a simple complemented ditopology they correspond for the most part to the point-free fuzzy separation axioms of B. Hutton and I. Reilly [5]. Hence our axioms generalize pointed versions of these separation axioms and show that, just as for classical general topology we can use the points or ignore them as best suits our purpose.

The notion of complete regularity is somewhat apart from the above as the definition involves an external space. We have found it convenient to approach the question of defining appropriate notions of complete regularity and complete coregularity via an extension of Urysohn's Lemma to ditopological texture spaces. For fuzzy topology Urysohn's lemma is usually stated in terms of the fuzzy unit interval, which has quite a complicated structure. For ditopological texture spaces, however, we may use the texture ([0, 1], L), where L={[0, r), [0, r] | r in [0, 1]} with the complement \sigma([0, r))=[0, 1-r], \sigma([0, r])=[0, 1-r), r in [0, 1], and ditopology \tau = {[0, r) | r in [0, 1]} \cup {[0, 1]}, \kappa = {[0, r] | r in [0, 1]} \cup {\emptyset}, which is in many ways an exact analogue of the unit interval with its usual topology. This space is not directly available in fuzzy topology, since it is not simple, but can be used to give a natural analogue of Urysohn's lemma for ditopological texture spaces, which we state in terms of difunctions [1], and the required concepts of complete regularity and complete coregularity. That these are indeed appropriate notions is further confirmed by Selma Özçag's work on uniform ditopologies, to be reported at this conference.

References

1. L. M. Brown, Relations and functions on texture spaces, Preprint.
2. L. M. Brown and R. Ertürk, Fuzzy Sets as Texture Spaces, I. Representation Theorems, Fuzzy Sets and Systems, 110 (2), 2000, 227-236.
3. L. M. Brown and R. Ertürk, Ditopological Texture Spaces and Fuzzy Topology, I. General Concepts, Submitted.
4. R. Ertürk, Separation axioms in fuzzy topology characterized by bitopologies, Fuzzy Sets and Systems, 58, 1993, 209-206.
5. B. Hutton and I. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy sets and Systems, 3, 1980, 93-104.

Date received: August 3, 2001