# Article

Full entry | PDF   (0.3 MB)
Keywords:
$\sigma$-frame; nearness; Samuel compactification
Summary:
We introduce the structure of a nearness on a $\sigma$-frame and construct the coreflection of the category ${\mathbf N\sigma Frm}$ of nearness $\sigma$-frames to the category ${\mathbf KReg\sigma Frm}$ of compact regular $\sigma$-frames. This description of the Samuel compactification of a nearness $\sigma$-frame is in analogy to the construction by Baboolal and Ori for nearness frames in [1] and that of Walters for uniform $\sigma$-frames in [11]. We also construct the uniform coreflection of a nearness $\sigma$-frame, that is, the coreflection of the category of ${\mathbf N\sigma Frm}$ to the category U$\sigma$Frm of uniform $\sigma$-frames.
References:
[1] D. Baboolal, R. G. Ori: Samuel Compactification and Uniform Coreflection of Nearness Frames. Proc. Symp. on Categorical Topology. University of Cape Town, Cape Town, 1994. MR 1722563
[2] B. Banaschewski: $\sigma$-Frames. Unpublished manuscript. Zbl 0675.06005
[3] B. Banaschewski: The frame envelope of a $\sigma$-frame. Quaest. Math. 16 (1993), 51–60. MR 1217474 | Zbl 0779.06009
[4] B. Banaschewski: Completion in pointfree topology. Lecture Notes in Math. and Applied Math, Univ. of Cape Town, SoCat 94, No2/1996. MR 1621835 | Zbl 1034.06008
[5] B. Banaschewski, C.  Gilmour: Stone-Čech compactification and dimension theory for regular $\sigma$-frames. J.  London Math. Soc. 39 (1989), 1–8. MR 0989914
[6] B. Banaschewski, A.  Pultr: Samuel compactification and completion of uniform frames. Math. Proc. Camb. Phil. Soc. 108 (1990), 63–78. DOI 10.1017/S030500410006895X | MR 1049760
[7] B. Banaschewski, A.  Pultr: Cauchy points of uniform and nearness frames. Quaest. Math. 19 (1996), 101–127. DOI 10.1080/16073606.1996.9631828 | MR 1390475
[8] P. T. Johnstone: Stone Spaces. Cambridge Studies in Advanced Math. No.  3. Cambridge University Press, Cambridge, 1982. MR 0698074
[9] J. J. Madden: $\kappa$-frames. J.  Pure Appl. Algebra 70 (1991), 107–127. MR 1100510 | Zbl 0721.06006
[10] I. Naidoo: Nearness and convergence in pointfree topology. PhD.  Thesis, University of Cape Town, Cape Town, 2004.
[11] J. L. Walters: Uniform sigma frames and the cozero part of uniform frames. Masters Dissertation, University of Cape Town, Cape Town, 1990.
[12] J. L. Walters: Compactifications and uniformities on $\sigma$-frames. Comm. Math. Univ. Carolinae 32 (1991), 189–198. MR 1118301 | Zbl 0735.54014
[13] J. L. Walters-Wayland: Completeness and nearly fine uniform frames. PhD.  Thesis, Univ. Catholique de Louvain, Louvain, 1996.

Partner of