On the Upper Completeness of Quasi-metric Spaces

This paper is concerned with the problem of upper completeness in the quasi-metric spaces. In this paper, firstly, some new basic concepts of quasi-metric spaces such as the upper limit and lower limit are put forward. Correspondingly, the concepts of upper closed set, upper Cauchy sequence and upper completeness are obtained. Secondly, three important examples in quasi-metric spaces are given; and some important results about them are attained. Thirdly, the conclusion that Hausdorff semi-distance space is an upper completeness quasi-metric space is received. Finally, the result about completeness of fractal spaces is extended into Hausdorff semi-metric spaces.