### Uniformities and covering properties for partial frames (I)

#### Abstract

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 suciently general to include as examples bounded distributive

lattices, -frames, -frames and frames.

Re

ective subcategories of uniform and nearness spaces and lately core-

ective subcategories of uniform and nearness frames have been a topic of

considerable interest. In [9] an easily implementable criterion for establishing

certain core

ections 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 unies 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 nite

meets distribute over these. After presenting our axiomatization of partial

frames, which we call S-frames, we add structure, in the form of S-covers

and nearness, and provide the promised method of constructing certain core-

ections. We illustrate the method with the examples of uniform, strong and

totally bounded nearness S-frames.

In Part (II) of this paper ([10]) we consider regularity, normality and

compactness for partial frames.

#### Keywords

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References

D. Baboolal and R.G. Ori, Samuel compactication and uniform core

ection of

nearness frames, Proceedings Symposium on Categorical Topology (1994), University

of Cape Town, 1999.

B. Banaschewski, Completion in pointfree topology, Lecture Notes in Math. and

Applied Math., University of Cape Town, No. 2 (1996).

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.

B. Banaschewski and C.R.A. Gilmour, Realcompactness and the cozero part of a

frame, Appl. Categ. Structures 9 (2001), 395-417.

B. Banaschewski, S.S. Hong, and A. Pultr, On the completion of nearness frames,

Quaest. Math. 21 (1998), 19-37.

B. Banaschewski and A. Pultr, A general view of approximation, Appl. Categ. Structures

(2006), 165-190.

B. Banaschewski and A. Pultr, Cauchy points of uniform and nearness frames,

Quaest. Math. 19 (1996), 101-127.

T. Dube, A note on complete regularity and normality, Quaest. Math. 19 (1996),

-478.

J. Frith and A. Schauerte, A method for constructing core

ections for nearness

frames, Appl. Categ. Structures (to appear).

J. Frith and A. Schauerte, Uniformities and covering properties for partial frames

(II), Categ. General Alg. Struct. Appl. 2(1) (2014), 23-35.

P.T. Johnstone, Stone Spaces", Cambridge University Press, Cambridge, 1982.

J.J. Madden, -frames, J. Pure Appl. Algebra 70 (1991), 107-127.

I. Naidoo, Aspects of nearness in -frames, Quaest. Math. 30 (2007), 133-145.

J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets

(Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 84-99.

J. Picado and A. Pultr, Frames and Locales", Springer, Basel, 2012.

Uniformities and covering properties for partial frames (I) 21

J. Picado, A. Pultr, and A. Tozzi, Locales, chapter in Categorical Foundations, MC

Pedicchio and W Tholen (eds), Encyclopedia of Mathematics and its Applications

, Cambridge University Press, Cambridge, (2004) 49-101.

S. Vickers, Topology via Logic", Cambridge Tracts in Theoretical Computer Science

, Cambridge University Press, Cambridge, 1989.

J. Walters, Compactications and uniformities on sigma frames, Comment. Math.

Univ. Carolinae 32(1) (1991), 189-198.

E.R. Zenk, Categories of partial frames, Algebra Universalis 54 (2005), 213-235.

D. Zhao, On projective Z-frames, Canad. Math. Bull. 40(1) (1997), 39-46.

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