Convergent net in a quasi-uniform space which is not Cauchy

Question

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a convergent net in a quasi-uniform space which is not Cauchy?

Answer 1

There are several definitions of Cauchy filters on a quasi-uniform space $(X,\mathcal U)$ [K]. For instance, a filter $\mathcal F$ on $(X,\mathcal U)$ is called

• a left $K$-Cauchy (resp. right $K$-Cauchy) filter, if for each $U\in\mathcal U$ there is $F\in\mathcal F$ such that $U(x)\in \mathcal F$ (resp. $U^{-1}(x)\in \mathcal F $) whenever $x\in F$;

• a $\mathcal U^*$-Cauchy filter, if for each $U\in\mathcal U$ there is $F\in\mathcal F$ such that $F\times F\subset U$;

• a $D$-Cauchy filter, if there exists a co-filter $\mathcal G$ on $X$ (that is, for each $U\in\mathcal U$ there are $F\in\mathcal F$ and $G\in\mathcal G$ such that $G\times F\subset U$;

• a PS (that is, Pervin-Sieber)-Cauchy filter, if for each $U\in\mathcal U$ there is $x\in X$ such that $U(x)\in\mathcal F$;

• a weakly Cauchy filter, if for each $U\in\mathcal U$, $\bigcap_{F\in\mathcal F} U^{-1}(F)\ne\varnothing$.

Convergent filters are not necessarily “each-Cauchy”. For instance, in general convergent filters are not left $K$-Cauchy: [K] e.g. a regular quasi-metric space in which each convergent sequence has a left $K$-Cauchy subsequence is metrizable (see [KMRV, Proposition 4]).

References

[K] H.P.A. Künzi, Quasi-uniform Spaces in the Year 2001 (in Recent Progress in General Topology II, pages 313-344; link to preprint).

[KMRV] H.P.A. Künzi, M. Mršević, I. L. Reilly, and M. K. Vamanamurthy, Convergence, precompactness and symmetry in quasi-uniform spaces, Math. Japonica 38 (1993), 239--253.