Cauchy Structures and Contiguities
by
Ákos Czászár

Let X be a set, 2 <= m in N, \Phi_m(X) denote the set of all subsets [m\tilde] of expX such that |[m\tilde]| <= m. An m-contiguity on X is a set [M\tilde] subset \Phi_m(X) such that

$M_1$ \emptyset in [m\tilde] in \Phi_m(X) implies [m\tilde] in [M\tilde],
$M_2$ [m\tilde] in [M\tilde] implies \cap [m\tilde] = \emptyset,
$M_3$ [m\tilde] in [M\tilde], [m\tilde] << [(m')\tilde] in \Phi_m(X) imply [(m')\tilde] in [M\tilde],
$M_4$ [m\tilde], [m\tilde]' in [M\tilde] implies [m\tilde] \/ [(m')\tilde] in [M\tilde] whenever [m\tilde] \/ [(m')\tilde] exists.

Here [m\tilde] << m^' iff M in [m\tilde] implies the existence of M' in [(m')\tilde] such that M contains M', and [m\tilde] \/ [(m')\tilde] exists iff [m\tilde] = {M₁, M₂, ... , M_s }, [(m')\tilde] = {M₁', M₂, ... , M_s }, and then [m\tilde] \/ [(m')\tilde] = {M₁ \cup M₁', M₂, ... , M_s }. For m=2, m-contiguities coincide with (Cech) proximities.

Any Cauchy structure on X induces an m-contiguity in a natural way. The purpose of the author is to investigste the question: which m-contiguities can be obtained from a Cauchy structure?

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-16.