# Uniformities and covering properties for partial frames (II)

Document Type: Research Paper

Authors

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

Abstract

This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which 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 there our axiomatization of partial frames, which we call $sels$-frames, we added structure, in the form of $sels$-covers and nearness.  Here, in the unstructured setting, we consider regularity, normality and compactness, expressing all these properties in terms of $sels$-covers. We see that an $sels$-frame is normal and regular if and only if the collection of all finite $sels$-covers forms a basis for an $sels$-uniformity on it. Various results about strong inclusions culminate in the proposition that every compact, regular $sels$-frame has a unique compatible $sels$-uniformity.

Keywords

### References

[1] B. Banaschewski, -frames, unpublished manuscript, 1980. Available online at http://mathcs.chapman.edu/CECAT/members/Banaschewski publications.
[2] J. Frith and A. Schauerte, Uniformities and covering properties for partial frames (I), Categ. General Alg. Struct. Appl. 2(1) (2014), 1-21.
[3] C.R.A. Gilmour, Realcompact spaces and regular -frames, Math. Proc. Camb. Phil. Soc. 96 (1984), 73-79.
[4] J.J. Madden, -frames, J. Pure Appl. Algebra 70 (1991), 107-127.
[5] J. Madden and J. Vermeer, Lindelof locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480.
[6] J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 84-99.
[7] J. Picado and A. Pultr, Frames and Locales", Springer, Basel, 2012.

[8] E.R. Zenk, Categories of partial frames, Algebra Universalis 54 (2005), 213-235.
[9] D. Zhao, On projective Z-frames, Canad. Math. Bull. 40(1) (1997), 39-46.