Another proof of Banaschewski's surjection theorem

Abstract

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.

Keywords

• frame (locale)
• sublocale
• uniform frame
• quasi-uniform frame
• uniform embedding
• complete uniform frame
• completion
• cauchy map
• cauchy filter
• cauchy complete