A textural view of the distinction between uniformities and quasi-uniformities

In this paper the authors consider the interplay between di-uniformities on a texture and a complementation on that texture. It is shown that with each di-uniformity corresponds a second di-uniformity, called its complement. Di-uniformities that coincide with their complement are then called complemented, and it is verified that the uniform ditopology of a complemented di-uniformity is a complemented ditopology. The connection with the uniform bicontinuity of the complement of a difunction is also considered.The relation between quasi-uniformities and uniformities on a set X in the classical sense is then investigated in the setting of di-uniformities on the complemented discrete texture on X. It is shown that di-uniformities on this discrete texture correspond in a one-to-one way with quasi-uniformities on X, a quasi-uniformity being a uniformity if and only if the corresponding di-uniformity is complemented. This shows that while the difference between quasi-uniformities and uniformities in the classical description is a question of symmetry, this becomes a matter of complementation in the di-uniform case.