Short notes on quasi-uniform spaces. IV: Cauchy type properties
A quasi-uniformity is said to be fully Cauchy if for any Cauchy filter pair there is a coarsest one among the Cauchy filter pairs coarser than it. It is observed that each locally quiet quasi-uniformity is fully Cauchy. Furthermore each fully Cauchy quasi-uniformity is Cauchy, where a quasi-uniformity is called Cauchy provided that \((f^{-1} \cap g^{-1}, f^1 \cap g^1)\) is Cauchy whenever \((f^{-1}, f)\), \((g^{-1}, g)\) are Cauchy filter pairs that meet (i.e. each member of \(f^i\) hits each member of \(g^i\) \((i \in \{-1, 1\})\). It is proved that any totally bounded Cauchy quasi-uniformity is a uniformity. A quasi-uniformity is said to be pointwise Cauchy if any Cauchy filter pair in \(^f {\mathcal C}\) (equivalently \(^p {\mathcal C})\) has a coarsest one among the Cauchy filter pairs coarser than it. The author notes that any weakly co-regular and any Cauchy quasi-uniformity is pointwise Cauchy. Furthermore the following statements are shown to hold for a pointwise Cauchy quasi-uniformity: If it is \(D\)-complete, then it is \(C\)-complete. If it is \(D\)-complete, then it is bicomplete. If it is point-symmetric and \(D\)-complete, then it is strongly co-stable. If it is strongly co-stable, then \(D\)-completeness is equivalent to bicompleteness. (A quasi-uniformity \({\mathcal U}\) is \(C\)-complete provided that each Cauchy filter pair is convergent and it is strongly co-stable provided that each co-\(D\)-Cauchy filter is stable with respect to the uniformity \({\mathcal U}^s\).) It is observed that any \(C\)-complete \(S_1\) quasi-uniformity is fully Cauchy and that the topology induced by a pointwise Cauchy quasi-uniformity is \(S_2\). Moreover the author calls a quasi-uniformity filter-symmetric provided that \((f^1, f^{-1})\) is a Cauchy pair whenever \((f^{-1}, f^1)\) is a Cauchy pair. Note that any proximally symmetric quasi-uniformity is filter-symmetric. It is verified that a quasi-uniformity is filter-symmetric iff it is quiet and doubly co-stable. Finally, any Cauchy bounded filter-symmetric quasi-uniformity is shown to be a uniformity.
Cauchy bounded filter-symmetric quasi-uniformity
scientific article; zbMATH DE number 878137
scientific article