Infima and complements in the lattice of quasi-uniformities
If \(({\mathcal U}_i)_{i\in I}\) is a family of quasi-uniformities on a set \(X\) such that every \({\mathcal U}_i\) has a certain property (P) it is quite natural to ask whether \(\inf({\mathcal U}_i)_{i\in I}\) and \(\sup({\mathcal U}_i)_{i\in I}\) also have property (P), where the infimum and the supremum is taken in the lattice \((q(X), \subseteq)\) of all quasi-uniformities on \(X\). It shown that if (P) is the property of being proximally symmetric, then \(\inf({\mathcal U}_i)_{i\in I}\) has (P). However, there exists a set \(X\) and proximally symmetric quasi-uniformities \({\mathcal U}_1\) and \({\mathcal U}_2\) on \(X\) such that \(\sup({\mathcal U}_1,{\mathcal U}_2)\) is not proximally symmetric. On the other hand, if (P) is the property of being transitive, then \(\sup({\mathcal U}_i)_{i\in I}\) has (P) while there exists a set \(X\) and transitive quasi-uniformities \({\mathcal U}_1\) and \({\mathcal U}_2\) on \(X\) such that \(\inf({\mathcal U}_1,{\mathcal U}_2)\) is not transitive. Additionally, it is shown that the Pervin quasi-uniformity of an irreducible \(T_1\)-space \(X\) does not have a complement in the lattice \((q(X), \subseteq)\) whereas the fine quasi-uniformity of each countable \(T_1\)-space \(X\) has a complement in \((q(X),\subseteq)\).
proximally symmetric quasi uniform space
scientific article; zbMATH DE number 5167611
scientific article