Another proof of Banaschewski's surjection theorem

Document Type : Research Paper


1 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.

2 Department of Mathematics University of Coimbra PORTUGAL

3 Department of Applied Mathematics and ITI, MFF, Charles University, Malostranske nam. 24, 11800 Praha 1, Czech Republic



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.


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