zbMATH — the first resource for mathematics

Stone-Čech compactification of locales. II. (English) Zbl 0549.54017
The existence of the Stone-Čech compactification of a topological space is equivalent, classically, to the prime ideal theorem, and hence only slightly weaker than the axiom of choice. In part I [Houston J. Math. 6, 301-312 (1980; Zbl 0473.54026)] the authors proved constructively the existence of the Stone-Čech compactification of a locale by lattice theoretic arguments. Independently, P. T. Johnstone established the same result by a constructive modification of the method of Tychonoff.
Among the many other ways of obtaining the Stone-Čech compactification of a space, the most significant one is that which describes it as the space of maximal ideals of its algebra of bounded continuous real-valued functions. The present paper presents the constructive analogue of this approach, based on a syntactic description of the locale of maximal ideals of this algebra introducted by the second author.
Reviewer: R.A.Alo

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54C40 Algebraic properties of function spaces in general topology
54H12 Topological lattices, etc. (topological aspects)
54A05 Topological spaces and generalizations (closure spaces, etc.)
06B10 Lattice ideals, congruence relations
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
Full Text: DOI
[1] Banaschewski, B., Untersuchungen über filterräume, ()
[2] Banaschewski, B.; Mulvey, G.J., Stone-čech compactification of locales, I. Houston J. math., 6, 301-312, (1980) · Zbl 0473.54026
[3] B. Banaschewski and C.J. Mulvey, Gelfand duality in a Grothendieck topos, to appear. · Zbl 1103.18001
[4] Fourman, M.P.; Grayson, R.J., Formal spaces, () · Zbl 0537.03040
[5] Fourman, M.P.; Hyland, J.M.E., Sheaf models for analysis, () · Zbl 0427.03028
[6] Johnstone, P.T., Tychonoff’s theorem without the axiom of choice, Fund. math., 113, 21-35, (1981) · Zbl 0503.54006
[7] Johnstone, P.T., Stone spaces, () · Zbl 0586.54001
[8] Joyal, A., Théorie des topos et le théorème de barr, Tagungsbericht of oberwolfach category meeting, (1977)
[9] Mulvey, C.J., A syntactic construction of the spectrum of a commutative C∗-algebra, Tagungsbereicht of oberwolfach category meeting, (1977)
[10] Tychonoff, A.N., Über die topologische erweiterung von Räumen, Math. ann., 102, 544-551, (1929) · JFM 55.0963.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.