Let $(X,\mathcal{V})$ is a uniform space and $\xi$ is a Cauchy filter on $(X,\mathcal{V})$.
$o(\xi)$ is the family of all open subsets of $X$ containing at least one element of $\xi$.
what does "containing at least one element of $\xi$" mean? that is, for every elements $V$ of $o(\xi)$ there exist at least one $A\in\xi$ such that $A\subset V$; or somthing else? Is $o(\xi)$ a Cauchy filter on $(X,\mathcal{V})$?
$U\in o(\xi)$ iff $U\subseteq X$ is open in the topology induced by $\mathcal V$ and there exists some $A\in \xi $ such that $A\subseteq U$. In general, $o(\xi)$ will not be a filter since there is no reason for it to have the property that if $U\subseteq V$ and $U\in o(\xi)$, then $V\in o(\xi)$ (as $V$ may easily fail to be open).