Di-Uniformities on Texture Spaces
by
Selma Ozcag
Hacettepe University, Ankara
Coauthors: Lawrence M. Brown (Hacettepe University)

By a texturing [1-4] of a set S we mean a subset S of the power set P(S) which is a point separating complete, completely distributive lattice with respect to inclusion which contains S and \emptyset, and for which arbitrary meets coincide with intersections and finite joins coincide with unions. The pair (S, S) is then called a texture space, or simply texture. Lattices of fuzzy sets may be represented by textures [3], which are proving a useful setting for the discussion of issues relating to symmetry and asymmetry.

In this paper we consider uniformities in a textural setting.

In the classical case the entourages of a diagonal uniformity on S are simply binary relations on the set S. The standard theory of relations is largely inappropriate for textures, however, because in general a texturing S need not be closed under set complementation. The recently developed concept of direlation [1] on the other hand provides appropriate analogues for reflexivity, inverse, symmetry and composition, and in terms of these a definition of direlational uniformity may be given which is formally the same as the classical definition. Likewise, working in terms of dicovers [2], we give an equivalent definition of dicover uniformity. We use the general term di-uniformity for such structures on (S, S).

Given a di-uniformity on the texture (S, S) we define the uniform ditopology generated by this di-uniformity, where a ditopology [1, 4] consists of possibly unrelated families of open sets and of closed sets. By an argument formally very similar to that used for uniformities [5], we show that the uniform ditopology satisfies the dual properties of complete regularity and complete coregularity [4].

In the direlational sense a di-uniformity is a symmetric structure, and indeed this hypothesis is essential in proving the above mentioned property of the uniform ditopology. On the other hand, when we specialise to the texture (S, P(S)) we see that a di-uniformity corresponds to a quasi-uniformity in the usual sense. We show that what distinguishes ``uniformity" from ``quasi-uniformity" in the direlational setting is not symmetry but a form of complementational invariance. Hence direlations give us a concept of symmetry which is essentially different from that for relations.

References

[1] Brown, L. M., Relations and functions on texture spaces, Preprint.

[2] Brown, L. M. and Diker, M., Paracompactness and full normality in ditopological texture spaces, J. Math. Anal. Appl., 227, 1998, 144-165.

[3] L. M. Brown and R. Ertürk, Fuzzy Sets as Texture Spaces, I. Representation Theorems, Fuzzy Sets and Systems, 110 (2), 2000, 227-236.

[4] L. M. Brown and R. Ertürk, Ditopological Texture Spaces and Fuzzy topology, II. Separation Axioms, Preprint.

[5] Kelley, J. L., General Topology, New York, 1955.

Date received: July 16, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagx-39.