Abstract
In this chapter we will discuss complete regularity. It is very naturally conservatively extended and hence the basic facts will be similar to those in classical spaces. In some respects, however, the point-free approach brings better results. Some of the facts are just more transparent, but there are also results that are entirely out of the classical scope. In particular, in the category of (completely regular) spaces there is no Lindelöf reflection; in the localic extension there is, and quite an interesting one.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It should be noted that the question can be simpler—although because of the algebraic nature of the approach geometrically less intuitive—in the point-free context, as discussed in [181].
- 2.
Here we use the Axiom of Countably Dependent Choice (CDC)—recall 1.1.1.
- 3.
Recall that CC is weaker than Countably Dependent Choice.
- 4.
The reader has certainly observed a similarity with the well-known classical fact on Lindelöf regular spaces being normal—see VII.1.5.1 below.
- 5.
And even more generally for normal regular frames.
References
B. Banaschewski. Untersuchungen über filterräume. Doctoral dissertation, Universität Hamburg, 1953.
B. Banaschewski. Another look at the localic Tychonoff theorem. Comment. Math. Univ. Carolinae 29 (1988) 647–656.
B. Banaschewski. The Real Numbers in Pointfree Topology. Textos de Matemática, vol. 12. University of Coimbra, 1997.
B. Banaschewski, C. J. Mulvey. Stone-Čech compactification of locales I. Houston J. Math. 6 (1980) 301–312.
B. Banaschewski, C. J. Mulvey. Stone-Čech compactification of locales II. J. Pure Appl. Algebra 33 (1984) 107–122.
B. Banaschewski, A. Pultr. A constructive view of complete regularity. Kyungpook Math. J. 43 (2003) 257–262.
C. H. Dowker, D. Strauss. Sums in the category of frames. Houston J. Math. 3 (1977) 7–15.
T. Dube, S. Iliadis, J. van Mill, I. Naidoo. A pseudocompact completely regular frame which is not spatial. Order 31 (2014) 115–120.
R. Engelking. General topology. Sigma Series in Pure Mathematics, vol. 6. Heldermann Verlag, Berlin, 1989.
L. Gillman, M. Jerison. Rings of continuous functions. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, 1960.
K. H. Hofmann, J. D. Lawson. The spectral theory of distributive continuous lattices. Trans. Amer. Math. Soc. 246 (1978) 285–310.
J. R. Isbell. Atomless parts of spaces. Math. Scand. 31 (1972) 5–32.
P. T. Johnstone. Tychonoff’s theorem without the axiom of choice. Fund. Math. 113 (1981) 21–35.
I. Kříž. A constructive proof of the Tychonoff’s theorem for locales. Comment. Math. Univ. Carolinae 26 (1985) 619–630.
I. Kříž. Factorization of frames and its application in “pointless topology”. Doctoral Dissertation, Charles University, 1988.
A. Mysior. A regular space which is not completely regular. Proc. Amer. Math. Soc. 81 (1981) 852–853.
J. Picado, A. Pultr. Frames and Locales: Topology without points. Frontiers in Mathematics, vol. 28, Springer, Basel, 2012.
G. Schlitt. The Lindelöf-Tychonoff theorem and choice principles. Math. Proc. Cambridge Phil. Soc. 110 (991) 57–65.
L. A. Steen, J. A. Seebach. Counterexamples in Topology. Springer-Verlag, 1970, 2nd ed. 1978.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Picado, J., Pultr, A. (2021). Complete Regularity. In: Separation in Point-Free Topology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-53479-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-53479-0_6
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-53478-3
Online ISBN: 978-3-030-53479-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)