Cover quasi-uniformities in frames
Quasi-uniformities (that is, not necessarily symmetric uniformities) on a set \(X\) are normally studied via entourages (that is, special subsets of \(X\times X\)). This has been done successfully in the point-free context, too. However, because the point-free representation of the square \(X\times X\) is not without difficulties, there was an interest in an approach to quasi-uniformities via covers. Based on the (spatial) ideas from Ganter and Steinlage, a cover-type quasi-uniformity was developed by Frith (and others) using biframes, the point-free variant of bitopologies. In the present paper, the authors show that this detour can be avoided by giving a variant of the paircover definition that avoids the need of a biframe structure. Indeed they present a concept of a cover-type quasi-uniformity structure that is directly defined in frames. Their new approach is then shown to be equivalent to the one based on the notion of a biframe.
scientific article; zbMATH DE number 5888192
scientific article