another proof of banaschewski’s surjection theorem

We present a new proof of banaschewski’s theorem stating that the completion lift of a uniform surjection is a surjection. the new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities (“not necessarily symmetric uniformities”). further, we show how a (regular) cauchy point on a closed uniform sublocale can be extended to a (regular) cauchy point on the larger (quasi-)uniform frame.