Convergence Structures in Numerical Analysis

Abstract

The paper deals — under the viewpoint of topology — with discrete Cauchy spaces, which are spaces where a discrete Cauchy structure $(t,\mathcal{C})$ (with $t$ being a discrete convergence and $\mathcal{C}$ being a discrete pre-Cauchy structure) is defined. More precisely, let $E_1,E_2,...$ and $E$ be arbitrary sets and let $\mathcal{S}$ denote the set of all discrete sequences $(x_n{,}_{n\in N^{\prime }}$ with $x_n,\in E_n(n\in N^{\prime })$ and with $N^{\prime }$ being an infinite subset of $\mathbb{N}=1,2,\dots$. Then $t$ and $\mathcal{C}$ are certain subsets of $(\mathcal{S},E)$ respectively of $\mathcal{S}$, which in a certain sense are assumed to be compatible. The paper gives properties of $t$ and $\mathcal{C}$ and among others is devoted to the problem of completion of discrete Cauchy spaces $(((E_1,E_2,\dots ),E);(t,\mathcal{C}))$. 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 $E$ is equipped with a metric $d$ and that there exist mappings $q_n:E_n\to E(n\in \mathbb{N})$ — the discrete Cauchy structure $(t,\mathcal{C})$ is defined by

It turns out that such a metric discrete Cauchy space is complete if and only if $(E,d)$ 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.