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Mapping a set of reals onto the reals

  • Arnold W. Miller (a1)
  • DOI:
  • Published online: 01 March 2014

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.

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[1]F. Bagemihl and H. Sprinkle , On a proposition of Sierpinski's which is equivalent to the continuum hypothesis, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 726728.

[2]J. Baumgartner and R. Laver , Iterated perfect set forcing, Annals of Mathematical Logic, vol. 17 (1979), pp. 271288.

[4]J. Isbell , A set whose square can map onto a perfect set, Proceedings of the American Mathematical Society, vol. 20 (1969), pp. 254255.

[7]R. Laver , On the consistency of Borel's conjecture. Acta Mathematica, vol. 137 (1976), pp. 151169.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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