Uniformities and covering properties for partial frames (I)

Document Type: Research Paper

Authors

Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.

Abstract

Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive lattices, $sigma$-frames, $kappa$-frames and frames.  Reflective subcategories of uniform and nearness spaces and lately coreflective subcategories of uniform and nearness frames have been a topic of considerable interest. In cite{jfas9} an easily implementable criterion for establishing certain coreflections in nearness frames was presented. Although the primary application in that paper was in the setting of nearness frames, it was observed there that similar techniques apply in many categories; we establish here, in this more general setting of structured partial frames, a technique that unifies these. We make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting our axiomatization of partial frames, which we call $sels$-frames, we add structure, in the form of $sels$-covers and nearness, and provide the promised method of constructing certain coreflections. We illustrate the method with the examples of uniform, strong and totally bounded nearness $sels$-frames.  In Part (II) of this paper, we consider regularity, normality and compactness for partial frames.

Keywords


[1] D. Baboolal and R.G. Ori, Samuel compacti cation and uniform coreection of nearness frames, Proceedings Symposium on Categorical Topology (1994), University of Cape Town, 1999.
[2] B. Banaschewski, Completion in pointfree topology, Lecture Notes in Math. and Applied Math., University of Cape Town, No. 2 (1996).
[3] B. Banaschewski, Uniform completion in pointfree topology, chapter in Topological and Algebraic Structures in Fuzzy Sets, S.E. Rodabaugh and E.P. Klement (Ed.s), Kluwer Academic Publishers, (2003) 19-56.
[4] B. Banaschewski and C.R.A. Gilmour, Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9 (2001), 395-417.
[5] B. Banaschewski, S.S. Hong, and A. Pultr, On the completion of nearness frames, Quaest. Math. 21 (1998), 19-37.
[6] B. Banaschewski and A. Pultr, A general view of approximation, Appl. Categ. Structures 14 (2006), 165-190.
[7] B. Banaschewski and A. Pultr, Cauchy points of uniform and nearness frames, Quaest. Math. 19 (1996), 101-127.
[8] T. Dube, A note on complete regularity and normality, Quaest. Math. 19 (1996), 467-478. [9] J. Frith and A. Schauerte, A method for constructing coreections for nearness frames, Appl. Categ. Structures (to appear).
[10] J. Frith and A. Schauerte, Uniformities and covering properties for partial frames (II), Categ. General Alg. Struct. Appl. 2(1) (2014), 23-35.
[11] P.T. Johnstone, Stone Spaces", Cambridge University Press, Cambridge, 1982.
[12] J.J. Madden, -frames, J. Pure Appl. Algebra 70 (1991), 107-127.
[13] I. Naidoo, Aspects of nearness in -frames, Quaest. Math. 30 (2007), 133-145.
[14] J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 84-99.
[15] J. Picado and A. Pultr, Frames and Locales", Springer, Basel, 2012.
[16] J. Picado, A. Pultr, and A. Tozzi, Locales, chapter in Categorical Foundations, MC Pedicchio and W Tholen (eds), Encyclopedia of Mathematics and its Applications 97, Cambridge University Press, Cambridge, (2004) 49-101.
[17] S. Vickers, Topology via Logic", Cambridge Tracts in Theoretical Computer Science
5, Cambridge University Press, Cambridge, 1989.
[18] J. Walters, Compacti cations and uniformities on sigma frames, Comment. Math.
Univ. Carolinae 32(1) (1991), 189-198.
[19] E.R. Zenk, Categories of partial frames, Algebra Universalis 54 (2005), 213-235.
[20] D. Zhao, On projective Z-frames, Canad. Math. Bull. 40(1) (1997), 39-46.