Short notes on quasi-uniform spaces. I: Uniform local symmetry (Q1912660)

The author calls a quasi-uniformity \({\mathcal U}\) uniformly locally symmetric provided that for any \(U \in {\mathcal U}\) there is a \(V \in {\mathcal U}\) such that for any \(x \in X\) there is a \(W \in {\mathcal U}\) with \(W^{-1} (V(x)) \subseteq U(x)\). Clearly each uniformly locally symmetric quasi-uniformity is locally symmetric. He proves that each mixed-symmetric, uniformly regular quasi-uniform space is uniformly locally symmetric and that each uniformly locally symmetric quasi-uniformity is quiet. (A quasi-uniformity is called quiet provided that the Cauchy filter pairs are uniformly weakly concentrated.) Furthermore he shows that a quasi-uniform space is uniformly locally symmetric provided that it has a uniformly locally symmetric sup-dense subspace. Some counterexamples are also constructed, e.g. an open-symmetric, not point-symmetric \(T_2\)-quasi-uniformity. The paper is well motivated by numerous additional interesting remarks on symmetry properties often studied in quasi-uniform spaces.