One-point compactifications and continuity for partial frames

Document Type : Research Paper


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


Locally compact Hausdorff spaces and their one-point compactifications are much used in topology and analysis; in lattice and domain theory, the notion of continuity captures the idea of local compactness. Our work is located in the setting of pointfree topology, where lattice-theoretic methods can be used to obtain topological results.
Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such.

Partial frames are meet-semilattices in which not all subsets need have joins.
A distinguishing feature  of their study is that a small collection  of axioms of an elementary nature allows one to do much that is  traditional for frames or locales. The axioms are sufficiently general to include as examples $\sigma$-frames, $\kappa$-frames and frames.

In this paper, we present the notion of a continuous partial frame by means of a suitable ``way-below'' relation; in the regular case this relation can be characterized using separating elements, thus avoiding any use of pseudocomplements (which need not exist in a partial frame). Our first main result is an explicit construction of a one-point compactification for a regular continuous partial frame using generators and relations.  We use strong inclusions to link continuity and one-point compactifications to least compactifications. As an application, we show that a one-point compactification of a zero-dimensional continuous partial frame is again zero-dimensional.

We next consider arbitrary compactifications of regular continuous partial frames. In full frames, the natural tools to use are right and left adjoints of frame maps; in partial frames these are, in general, not available. This necessitates significantly different techniques to obtain largest and smallest elements of fibres (which we call balloons); these elements are then used to investigate the structure of the compactifications. We note that strongly regular ideals play an important r\^{o}le here. The paper concludes with a proof of the uniqueness of the one-point compactification.


Dedicated  to Bernhard Banaschewski on the occasion of his $90^{th}$ birthday


[1] Adamek, J., Herrlich, H., and Strecker, G., "Abstract and Concrete Categories", John Wiley & Sons Inc., New York, 1990.
[2] Baboolal, D., Conditions under which the least compactification of a regular continuous frame is perfect, Czechoslovak Math. J. 62(137) (2012), 505-515.
[3] Baboolal, D., N-star compactifications of frames, Topology Appl. 168 (2014), 8-15.
[4] Banaschewski, B., The duality of distributive $sigma$-continuous lattices, in: "Continuous lattices", Lecture Notes in Math. 871 (1981), 12-19.
[5] Banaschewski, B., Compactification of frames, Math. Nachr. 149 (1990), 105-116.
[6] Banaschewski, B., $sigma$-frames, unpublished manuscript, 1980. Available at publications
[7] Banaschewski B. and Gilmour, C.R.A., Stone-Cech compactification and dimension theory for regular $sigma$-Frames, J. London Math. Soc. 39(2) (1989), 1-8.
[8] Banaschewski, B. and Gilmour, C.R.A., Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9 (2001), 395-417.
[9] Banaschewski, B. and Gilmour, C.R.A., Cozero bases of frames, J. Pure Appl. Algebra 157 (2001), 1-22.
[10] Banaschewski, B. and Matutu, P., Remarks on the frame envelope of a $sigma$-frame, J. Pure Appl. Algebra 177(3) (2003), 231-236.
[11] Erne, M. and Zhao, D., Z-join spectra of Z-supercompactly generated lattices, Appl. Categ. Structures 9(1) (2001), 41-63.
[12] Frith, J. and Schauerte, A., The Samuel compactification for quasi-uniform biframes, Topology Appl. 156 (2009), 2116-2122.
[13] Frith, J. and Schauerte, A., Uniformities and covering properties for partial frames (I), Categ. General Alg. Struct. Appl. 2(1) (2014), 1-21.
[14] Frith, J. and Schauerte, A., Uniformities and covering properties for partial frames (II), Categ. General Alg. Struct. Appl. 2(1) (2014), 23-35.
[15] Frith, J. and Schauerte, A., The Stone-Cech compactification of a partial frame via ideals and cozero elements, Quaest. Math. 39(1) (2016), 115-134.
[16] Frith, J. and Schauerte, A., Completions of uniform partial frames, Acta Math. Hungar. 147(1) (2015), 116-134.
[17] Frith, J. and Schauerte, A., Coverages give free constructions for partial frames, Appl. Categ. Structures, Available online (2015), DOI: 10.1007/s10485-015-9417-8
[18] Frith, J. and Schauerte, A., Compactifications of partial frames via strongly regular ideals, Math. Slovaca, accepted June 2016.
[19] Gutierrez Garcia, J., Mozo Carollo, I., and Picado, J., Presenting the frame of the unit circle, J. Pure and Appl. Algebra 220(3) (2016), 976-1001.
[20] Johnstone, P.T., "Stone Spaces", Cambridge University Press, 1982.
[21] Lee, S.O., Countably approximating frames, Commun. Korean Math. Soc. 17(2) (2002), 295-308.
[22] Mac Lane, S., "Categories for the Working Mathematician", Springer-Verlag, 1971.
[23] Madden, J.J., -frames, J. Pure Appl Algebra 70 (1991), 107-127.
[24] Paseka, J., Covers in generalized frames, in: "General Algebra and Ordered Sets" (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 84-99.
[25] Paseka, J. and Smarda, B., On some notions related to compactness for locales, Acta Univ. Carolin. Math. Phys. 29(2) (1988), 51-65.
[26] Picado, J. and Pultr, A., "Frames and Locales", Springer, 2012.
[27] Walters, J.L., Compactifications and uniformities on $sigma$-frames, Comment. Math. Univ. Carolin. 32(1) (1991), 189-198.
[28] Zenk, E.R., Categories of partial frames, Algebra Universalis 54 (2005), 213-235.
[29] Zhao, D., Nuclei on Z-frames, Soochow J. Math. 22(1) (1996), 59-74.
[30] Zhao, D., On Projective Z-frames, Canad. Math. Bull. 40(1) (1997), 39-46.
[31] Zhao, D., Closure spaces and completions of posets, Semigroup Forum 90(2) (2015), 545-555.