\(L\)-proximities and totally bounded pointwise \(L\)-uniformities
The connections between \(L\)-(quasi-)proximities in the sense of Katsaras-Liu-Artico-Moresco and pointwise \(L\)-(quasi-)uniformities in the sense of Shi are investigated. A pointwise (quasi-) uniform space \((L^x,{\mathcal U})\) and an \(L\)-(quasi-)proximity space \((L^x,\delta)\) are considered. For an \(L\)-proximity \(\delta\) on \(L^x\), a pointwise \(L\)-quasi-uniformity \({\mathcal U}_\delta\) is defined. It is proved that \(\delta\) and \({\mathcal U}_\delta\) induce the same topology. An \(L\)-topology can be induced by an \(L\)-proximity iff it is pointwise completely regular. For any \(L\)-(quasi-)proximity \(\delta\), the \(L\)-(quasi-)uniformity \({\mathcal U}_\delta\) is totally bounded.
scientific article; zbMATH DE number 1868238