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A very discontinuous Borel function

Published online by Cambridge University Press:  12 March 2014

Juris Steprāns
Juris Steprāns*
Affiliation:
Department of Mathematics, York University, North York, Ontario, CanadaM3J IP3
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Abstract

It is shown to be consistent that the reals are covered by ℵ1, meagre sets yet there is a Baire class 1 function which cannot be covered by fewer than ℵ2, continuous functions. A new cardinal invariant is introduced which corresponds to the least number of continuous functions required to cover a given function. This is characterized combinatorially. A forcing notion similar to, but not equivalent to, superperfect forcing is introduced.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 4 , December 1993 , pp. 1268 - 1283
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Abraham, A., Rubin, M., and Shelah, S., On the consistency of some partition theorems for continuous colorings, and structure of ℵ1-dense real order types, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 123–206.CrossRefGoogle Scholar
[2]Baumgartner, J., Iterated forcing, Surveys in set theory (Cambridge) (Mathias, A. D. R., editor), London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, London and New York, 1983, pp. 1–59.Google Scholar
[3]Blass, A., Applications of superperfect forcing and its relatives, Set theory and its applications (Berlin) (Steprāns, J. and Watson, S., editors), Lecture Notes in Mathematics, vol. 140, Springer-Verlag, Berlin and New York, 1989, pp. 18–40.CrossRefGoogle Scholar
[4]Cichoń, J., Morayne, M., Pawlikowski, J., and Solecki, S., Decomposing baire functions, this Journal, vol. 56 (1991), pp. 1273–1283.Google Scholar
[5]Engelking, R., General topology, Polish Scientific Publishers, Warsaw, 1977.Google Scholar
[6]Laver, R., On the consistency of Borel's conjecture, Acta Mathematica, vol. 137 (1976), pp. 151–169.CrossRefGoogle Scholar
[7]Miller, A., Rational perfect set forcing, Axiomatic set theory (Martin, D. A., Baumgartner, J., and Shelah, S., editors), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, pp. 143–159.CrossRefGoogle Scholar
[8]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[9]Todorčević, S., Remarks on chain conditions in products, Compositio Mathematica, vol. 55 (1985), pp. 295–302.Google Scholar
[10]Velickovic, B., OCA and automorphisms of P(ω)/fin, preprint, 1990.Google Scholar