Abstract
Of the classical separation axioms, normality is the easiest to extend. There is, basically, nothing “pointy” about it. This however does not mean that there is not much interest about it in the extended context. On the contrary, besides the new view one gains of the plain normality itself and of its relations to the other axioms one has natural strengthenings (and, in a smaller extent also weakenings) that are not so obviously point-free and the behaviour of which is of an independent interest.
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Notes
- 1.
A frame is almost normal whenever the (norm) condition holds for elements a, b at least one of which is regular (recall that an element x of a frame is regular if x ∗∗ = x).
- 2.
Closed localic maps are the localic maps corresponding to closed continuous maps—see, e.g., [220].
- 3.
The reader will certainly notice that we use the density of f only; but closed dense maps are onto anyway.
- 4.
Recall V.4.1.1. The fact that s is a codense homomorphism means that the sublocale L s is a codense sublocale of L, that is, (L s)∘ = L for the fitting operator
$$\displaystyle \begin{aligned}S^\circ=\mathop{\textstyle \bigcap}\{\mathfrak{o}(a)\mid S\subseteq \mathfrak{o}(a)\}\end{aligned}$$on sublocales [75, 2.5]. See also Chap. X.
References
B. Banaschewski, R. Harting. Lattice aspects of radical ideals and choice principles. Proc. London Math. Soc. (3) 50 (1985) 385–404.
M. M. Clementino, J. Picado, A. Pultr. The other closure and complete sublocales. Appl. Categ. Structures 26 (2018) 891–906; errata ibid. 26 (2018) 907–908.
C. H. Dowker, D. Strauss. Paracompact frames and closed maps. In: Symposia Mathematica, XVI, pp. 93–116. Convegno sulla Topologia Insiemistica e Generale, INDAM, Rome, 1973. Academic Press, London, 1975.
M. J. Ferreira. J. Gutiérrez García, J. Picado. Completely normal frames and real-valued functions. Topology Appl. 156 (2009) 2932–2941.
F. Hausdorff. Gestufte Räume. Fund. Math. 25 (1935) 486–502.
J. R. Isbell. Graduation and dimension in locales. In: Aspects of Topology (ed. by I. M. James, E. H. Kronheimer), pp. 195–210. London Math. Soc. Lecture Note Ser., vol. 93. Cambridge Univ. Press, Cambridge, 1985.
P. T. Johnstone. Wallman compactification of locales. Houston J. Math. 10 (1984) 201–206.
T. Kaiser. A sufficient condition of full normality. Comment. Math. Univ. Carolinae 37 (1996) 381–389.
J. Picado, A. Pultr. Frames and Locales: Topology without points. Frontiers in Mathematics, vol. 28, Springer, Basel, 2012.
A. Pultr, J. Úlehla. Notes on characterization of paracompact frames. Comment. Math. Univ. Carolinae 30 (1989) 377–384.
N. A. Shanin. On the theory of bicompact extensions of topological spaces. C. R. (Dokl.) Acad. Sci. URSS (N.S.) 38 (1943) 154–156.
H. Simmons. The lattice theoretic part of topological separation properties. Proc. Edinburgh Math. Soc. 21 (1978) 41–48.
S. H. Sun. On paracompact locales and metric locales. Comment. Math. Univ. Carolinae 30 (1989) 101–107; errata ibid. 30 (1989) 817–818.
H. Wallman. Lattices and topological spaces. Ann. Math. 39 (1938) 112–126.
G. T. Whyburn. Open and closed mappings. Duke Math. J. 17 (1950) 69–74.
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Picado, J., Pultr, A. (2021). Normality. In: Separation in Point-Free Topology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-53479-0_7
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