A class of uniform convergence structures

Abstract:

In 1967, Cook and Fischer introduced in the journal Mathematische Annalen the notion of a uniform convergence structure, abbreviated u.c.s., for a set $X$. Here we consider the class $\Gamma$ of u.c.s. which have the following property: a u.c.s. $I \in \Gamma$ provided there is a filter $\Phi \in I$ such that $\mathcal {F}$ is finer than $\Phi (x)$ for every filter $\mathcal {F}$ which converges to $x$, for each $x \in X$. Various properties of the class $\Gamma$ are discussed. The main result is that a topology $\tau$ for $X$ is regular if and only if there is an $I \in \Gamma$ such that $I$ induces $\tau$. Also it it is shown that each $I \in \Gamma$ induces a regular topology for $X$. The class ${\Gamma _0}$ of u.c.s. which satisfy the completion axiom was first introduced by Biesterfeldt, Indag. Math., 1966. Here it is shown that ${\Gamma _0} \subset \Gamma$ and a characterization of the class ${\Gamma _0}$ is given in terms of Cauchy filters.