Another proof of Banaschewski's surjection theorem

Document Type: Research Paper

Authors

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

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


[1] Banaschewski, B., Uniform completion in pointfree topology, In: Rodabaugh S.E., Klement E.P. (eds) Topological and Algebraic Structures in Fuzzy Sets, 19-56. Trends Log. Stud. Log. Libr. 20, Springer, 2003.
[2] Banaschewski, B., Hong, S.S., and Pultr, A., On the completion of nearness frames, Quaest. Math. 21 (1998), 19-37.
[3] Banaschewski, B. and Pultr, A., Samuel compactification and completion of uniform frames, Math. Proc. Cambridge Phil. Soc. 108 (1990), 63-78.
[4] Banaschewski, B. and Pultr, A., Cauchy points of uniform and nearness frames, Quaest. Math. 19 (1996), 101-127.
[5] Isbell, J.R., “Uniform Spaces”, Amer. Math. Soc., 1964.
[6] Johnstone, P.T., “Stone Spaces”, Cambridge University Press, 1982.
[7] Kelley, J.L., “General Topology”, D. Van Nostrand Co., 1955.
[8] KríΕΎ, I., A direct description of uniform completion in locales and a characterization of LT-groups, Cah. Topol. Géom. Différ. Catég. 27 (1986), 19-34.
[9] Picado, J., Weil uniformities for frames, Comment. Math. Univ. Carolin. 36 (1995), 357-370.
[10] Picado, J., Structured frames by Weil entourages, Appl. Categ. Structures 8 (2000), 351-366.
[11] Picado, J. and Pultr, A., “Frames and locales: Topology without points”, Frontiers in Mathematics 28, Springer, 2012.
[12] Picado, J. and Pultr, A., Entourages, covers and localic groups, Appl. Categ. Structures 21 (2013), 49-66.
[13] Picado, J. and Pultr, A., Entourages, density, Cauchy maps, and completion, Appl. Categ. Structures (2018), https://doi.org/10.1007/s10485-018-9542-2.
[14] Pultr, A., Pointless uniformities I, Comment. Math. Univ. Carolin. 25 (1984), 91-104.