Akos Csaszar
Cauchy Structures and Covering Structures
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 52, 3-10 (1998)

MSC:

54E17 Nearness spaces

Abstract: Let $X$ be a set. A $Cauchy\ structure\ \frak S$ on $X$ is a non-empty
collection of filters in $X$ such that
\begin{equation}\tag{0.1}
\overset{\cdot}{x}\in{\frak S}\text{ for } x\in X,
\end{equation}
\begin{equation}\tag{0.2}
\frak s \in \frak S\text{ and }\frak s\subset\frak s' \in \text{Fil }
X\text{ imply }\frak s' \in\frak S,
\end{equation}
\begin{equation}\tag{0.3}
\frak s, \frak s'\in\frak S \text{ and }\frak s\Delta\frak s'\in\frak S.
\end{equation}
Here $\overset{\cdot}{x}=\{ S\subset X: x\in S\}$, Fil $X$ is the
collection of all filters (proper or not) in $X$, and $\frak s\Delta\frak
s'$ iff $S\in\frak s,\ S' \in\frak s'$ imply $S\subset S'\neq\emptyset$
(see e.g. [3]). If $\frak S$ satisfies (0.1) and (0.2), it is said to be a
{\it screen} on $X$ (see e.g. [2]).
Keywords: Nearness spaces