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Mapping a set of reals onto the reals

Published online by Cambridge University Press:  12 March 2014

Arnold W. Miller
Arnold W. Miller*
Affiliation:
University of Texas, Austin, Texas 78712
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Abstract

In this paper we show that it is consistent with ZFC that for any set of reals of cardinality the continuum, there is a continuous map from that set onto the closed unit interval. In fact, this holds in the iterated perfect set model. We also show that in this model every set of reals which is always of first category has cardinality less than or equal to ω1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 575 - 584
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[1]Bagemihl, F. and Sprinkle, H., On a proposition of Sierpinski's which is equivalent to the continuum hypothesis, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 726–728.Google Scholar
[2]Baumgartner, J. and Laver, R., Iterated perfect set forcing, Annals of Mathematical Logic, vol. 17 (1979), pp. 271–288.CrossRefGoogle Scholar
[3]Isbell, J., Spaces without large projective subspaces, Mathematica Scandinarica, vol. 17 (1965), pp. 89–105.CrossRefGoogle Scholar
[4]Isbell, J., A set whose square can map onto a perfect set, Proceedings of the American Mathematical Society, vol. 20 (1969), pp. 254–255.CrossRefGoogle Scholar
[5]Kunen, K., Doctoral Dissertation, Stanford University, 1968.Google Scholar
[6]Kuratowski, K., Topology, vol. 1, Academic Press, New York, 1966.Google Scholar
[7]Laver, R., On the consistency of Borel's conjecture. Acta Mathematica, vol. 137 (1976), pp. 151–169.CrossRefGoogle Scholar
[8]Royden, H.L., Real analysis, Macmillan, New York, 1968.Google Scholar
[9]Sierpinski, W., Sur un ensemble non dénombrable donte toute image continue est de mesure null, Fundamenta Mathematical, vol. 11 (1928), p. 304.Google Scholar
[10]Sierpinski, W., Hypothèse du continu, Monografie Matematyczne, Tom IV, Warsaw, 1934.Google Scholar