Abstract
A classical result in the theory of uniform spaces is that any topological space with a base of clopen sets admits a uniformity with a transitive base and the uniform topology of such a space has a base of clopen sets. This paper presents a pointfree generalization of this, both to uniform and quasi-uniform frames, together with various properties concerning total boundedness, compactifications and completions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banaschewski, B.: Ñber nulldimensionale Räume, Math. Nachr. 13 (1955), 129–140.
Banaschewski, B.: Universal zero-dimensional compactifications, in Proc. Int. Conf. on Categorical Topology, Prague 1988, World Sci. Publishing, Singapore, 1989, pp. 257–269.
Banaschewski, B.: Compactifications of frames, Math. Nachr. 149 (1990), 105–116.
Banaschewski, B., Brümmer, G. C. L. and Hardie, K. A.: Biframes and bispaces, Quaestiones Math. 6 (1983), 13–25.
Banaschewski, B. and Pultr, A.: Samuel compactification and completion of uniform frames, Math. Proc. Cambridge Philos. Soc.108 (1990), 63–78.
Fletcher, P.: On completeness of quasi-uniform spaces, Arch. Math. 22 (1971), 200–204.
Fletcher, P. and Hunsaker, W.: Entourage uniformities for frames, Monatsh. Math. 112 (1991), 271–279.
Fletcher, P., Hunsaker, W. and Lindgren, W.: Frame quasi-uniformities, Monatsh. Math. 117 (1994), 223–236.
Fletcher, P., Hunsaker, W. and Lindgren, W.: Characterizations of frame quasi-uniformities, Quaestiones Math.16 (1993), 371–383.
Fletcher, P. and Lindgren, W.: Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.
Frith, J. L.: The category of quasi-uniform frames, Preprint, University of Cape Town, 1991.
Frith, J. L. and Hunsaker, W.: Completion of quasi-uniform frames, Appl. Categ. Structures 7 (1999), 261–270.
Hunsaker, W. and Picado, J.: A note on total boundedness, Preprint, 1998.
Isbell, J. R.: On zero-dimensional spaces, Tohuku Math. J. 7 (1955), 1–8.
Isbell, J. R.: Atomless parts of spaces, Math. Scand. 31 (1972), 5–32.
Johnstone, P. T.: Stone Spaces, Cambridge University Press, Cambridge, 1982.
Kelly, J. C.: Bitopological spaces, Proc. London Math. Soc. 13 (1963), 71–89.
Nyikos, P. and Reichel, H. C.: On the structure of zero-dimensional spaces, Indag. Math. 37 (1975), 120–136.
Pervin, W. J.: Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316–317.
Picado, J.: Weil uniformities for frames, Comment. Math. Univ. Carolin. 36 (1995), 357–370.
Picado, J.: Weil entourages in pointfree topology, Ph.D. Thesis, University of Coimbra, 1995.
Picado, J.: Frame quasi-uniformities by entourages, in Festschrift for G. C. L. Brümmer on his 60th Birthday, Cape Town 1994, Department of Mathematics, University of Cape Town, 1999, pp. 161–175.
Reilly, I. L.: Zero-dimensional bitopological spaces, Indag. Math. 35 (1973), 127–131.
Schauerte, A.: Biframe compactifications, Comment. Math. Univ. Carolin. 34 (1993), 567–574.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hunsaker, W., Picado, J. Frames with Transitive Structures. Applied Categorical Structures 10, 63–79 (2002). https://doi.org/10.1023/A:1013372903430
Issue Date:
DOI: https://doi.org/10.1023/A:1013372903430