The paper deals — under the viewpoint of topology — with discrete Cauchy spaces, which are spaces where a discrete Cauchy structure (with being a discrete convergence and being a discrete pre-Cauchy structure) is defined. More precisely, let and be arbitrary sets and let denote the set of all discrete sequences with and with being an infinite subset of . Then and are certain subsets of respectively of , which in a certain sense are assumed to be compatible. The paper gives properties of and and among others is devoted to the problem of completion of discrete Cauchy spaces . The construction of a completion of a discrete Cauchy space differs (in some sense essentially) from the construction of a completion of a usual sequential Cauchy space and is even more simple. An essential part of the paper is devoted to certain metric discrete Cauchy spaces, where — among others assuming that is equipped with a metric and that there exist mappings — the discrete Cauchy structure is defined by
It turns out that such a metric discrete Cauchy space is complete if and only if is complete and that also the completion is metric. A further subject of the paper are metric discrete Cauchy spaces of mappings between metric discrete Cauchy spaces, where simple characterizations of the corresponding discrete convergence and discrete pre-Cauchy structure of such a discrete Cauchy space as well as a necessary and sufficient condition for its completeness are given.
Cite this article
Siegfried Gähler, D. Matel-Kaminska, Convergence Structures in Numerical Analysis. Z. Anal. Anwend. 15 (1996), no. 2, pp. 529–544DOI 10.4171/ZAA/713