Cauchy filter

Remarks

The neighbourhood filter $\mathcal{N}_x$ of a point $x$ is a Cauchy filter. Given an entourage $U$, choose a symmetric entourage $V$ with $V \circ V \subseteq U$ and let $A = V[x]$, a member of $\mathcal{N}_x$. For $y, z \in A$ one has $(y, x) \in V$ and $(x, z) \in V$, so $(y, z) \in V \circ V \subseteq U$. On $\R$ with its usual uniformity, the filter of sets that contain some interval $[n, \infty)$ is not a Cauchy filter. For $U = \set{(x, y) : \abs{x - y} < 1}$, every member contains two points at distance $1$, so no member $A$ has $A \times A \subseteq U$. In a metric space, a sequence is a Cauchy sequence if and only if the filter of its tails is a Cauchy filter. Every convergent filter is a Cauchy filter.