Completeness types for uniformity theory on textures
In this paper the author considers the various types of completeness for di-uniform texture spaces and especially for complemented ones. Following that, the relationships between completeness of uniform spaces and these types of completeness for complemented di-uniform texture spaces are investigated in a categorical setting, just as interrelations between quasi-uniform spaces and di-uniform texture spaces are pointed out insofar as completeness is concerned. Additionally, useful requirements
... eful requirements among the various types of completeness of a di-uniformity and real dicompactness of the uniform ditopological space generated by that di-uniformity are presented as a diagram. The foundations of a suitable uniformity theory on textures giving descriptions in terms of direlations, dicovers and dimetrics have been developed in  and the term di-uniformity was introduced to cover both dicovering and direlational uniformities. In this work, constant reference will be made to  for definitions and results relating to di-uniformities, most of which will be repeated here. Following this, the relationships between quasi-uniformities and uniformities (see ) on a set in the classical sense are then investigated in  , in the setting of di-uniformities on a special texture. On the other hand, the subjects of completeness and total boundedness for di-uniformities are discussed in  and the term dicompleteness is defined as a type of completeness for di-uniform texture spaces. In addition,  gives a categorical point of view for the di-uniform texture spaces by defining various categories and functors. Motivation and background material specific to the main topic of this paper maybe found in  ,                . Due to lack of space, most of this material is not repeated here. In particular, the reader is referred to 2010 Mathematics Subject Classification. Primary 54E15, 54E50, 54B30, 54D35, 18A22 ; Secondary 54C30, 06D10, 03E20.