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The Lindelöf Tychonoff Theorem and choice principles

Published online by Cambridge University Press:  24 October 2008

G. Schlitt
G. Schlitt
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, B.C. V5A 1S6, Canada
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Abstract

It is an important result in frame theory that the coproduct of a family of regular Lindelöf frames is Lindelöf [3]. We show that this ‘Lindelöf Tychonoff Theorem’ or ‘LTT’ is independent of ZF and indeed lies close in logical strength to the Axiom of Countable Choice, quite unlike the case with the usual (frame) Tychonoff Theorem. Along the way we construct the regular Lindelöf coreflection and obtain a simple proof of the LTT as a corollary.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 57 - 65
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

‘1’Banaschewski, B.. Universal 0-dimensional Compactifications. In Categorical Topology and its Relation to Algebra, Analysis and Combinatorics (World Scientific, 1989), pp. 257–269.Google Scholar
‘2’Banaschewski, B. and Mulvey, C.. Stone–Cěch compactification of locales I. Houston J. Math. 6 (1980), 301–311.Google Scholar
‘3’Dowker, C. H. and Strauss, D.. Sums in the category of frames. Houston J. Math. 3 (1976), 17–32.Google Scholar
‘4’Johnstone, P. T.. The point of pointless topology. Bull. Amer. Math. Soc. 8 (1983), 41–53.CrossRefGoogle Scholar
‘5’Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. no. 3. (Cambridge University Press, 1982).Google Scholar
‘6’Johnstone, P. T.. Tychonoff's theorem without the axiom of choice. Fund. Math. 113 (1981), 31–35.Google Scholar
‘7’Jech, T.. The Axiom of Choice (North-Holland, 1973).Google Scholar
‘8’Kelley, J. L.. The Tychonoff product theorem implies the axiom of choice. Fund. Math. 37 (1950), 75–76.CrossRefGoogle Scholar
‘9’MacLane, S.. Categories for the Working Mathematician. Graduate Texts in Math. no. 5 (Springer-Verlag, 1971).Google Scholar
‘10’Madden, J. and Vermeer, J.. Lindelöf Locales and realcompactness. Math. Proc. Cambridge Philos. Soc. 99 (1986), 473–480.CrossRefGoogle Scholar
‘11’Schlitt, G.. N-compact frames (to appear).Google Scholar
‘12’Steen, L. A. and Seebach, J. A.. Counterexamples in Topology (Springer-Verlag, 1978).CrossRefGoogle Scholar
‘13’Vermeulen, J. J. C.. Constructive techniques in functional analysis. Ph.D. thesis, University of Sussex (1987).Google Scholar