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The Lindelöf Tychonoff theorem and choice principles. (English) Zbl 0737.03024

This paper comes as a rude shock to those (including the reviewer) who have for years maintained that locale theory is an “inherently constructive” way of doing general topology, and in particular that it avoids most of the uses of the axiom of choice (or of the Boolean prime ideal theorem) in traditional point-set topology. A prime example of this is the result, first discovered by the reviewer, that the Tychonoff theorem for locales (i.e. the assertion that a product of compact locales is compact) is provable in \(ZF\) set theory without choice; J. J. C. Vermeulen and others have since shown that the proof can even be made constructive. However, in the present paper, the author shows that the corresponding assertion for Lindelöf locales, which is provable in \(ZF\) plus countable choice, is not provable in \(ZF\); indeed, there are closely-related assertions about locales which are equivalent in \(ZF\) to countable choice. {Perhaps there is some consolation to be derived from the fact that the “Lindelöf Tychonoff theorem” is not true at all for spaces, even with the full axiom of choice.}.

MSC:

03E25 Axiom of choice and related propositions
06D20 Heyting algebras (lattice-theoretic aspects)
54B10 Product spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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References:

[1] DOI: 10.1090/S0273-0979-1983-15080-2 · Zbl 0499.54002 · doi:10.1090/S0273-0979-1983-15080-2
[2] Dowker, Houston J. Math. 3 pp 17– (1976)
[3] Banaschewski, Houston J. Math. 6 pp 301– (1980)
[4] Banaschewski, Categorical Topology and its Relation to Algebra, Analysis and Combinatorics pp 257– (1989)
[5] Steen, Counterexamples in Topology (1978) · doi:10.1007/978-1-4612-6290-9
[6] Johnstone, Stone Spaces (1982)
[7] Madden, Math. Proc. Cambridge Philos. Soc. 99 pp 473– (1986)
[8] Kelley, Fund. Math. 37 pp 75– (1950)
[9] Jech, The Axiom of Choice (1973)
[10] Johnstone, Fund. Math. 113 pp 31– (1981)
[11] MacLane, Categories for the Working Mathematician (1971)
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