Cover quasi-uniformities in frames

Quasi-uniformities (not necessarily symmetric uniformities) are usually studied via entourages (special neighbourhoods of the diagonal in X × X ) where one can simply forget about the symmetry requirement. This has been done successfully in the point-free context as well, but there is a demand for a covering approach, a.o. because the point-free representation of the square X × X is not without difficulties. Based on the (spatial) ideas from Gantner and Steinlage (1972) [9], a cover type quasi-uniformity was developed in Frith (1987) [6] and other papers using biframes, the point-free variant of bitopologies. In this paper we show that this can be avoided and present a cover type quasi-uniformity structure enriching that of frame directly.