DISTINGUISHING PERFECT SET PROPERTIES IN SEPARABLE METRIZABLE SPACES

@article{Medini2016DISTINGUISHINGPS,
  title={DISTINGUISHING PERFECT SET PROPERTIES IN SEPARABLE METRIZABLE SPACES},
  author={Andrea Medini},
  journal={The Journal of Symbolic Logic},
  year={2016},
  volume={81},
  pages={166 - 180}
}
  • Andrea Medini
  • Published 1 May 2014
  • Mathematics
  • The Journal of Symbolic Logic
Abstract All spaces are assumed to be separable and metrizable. Our main result is that the statement “For every space X, every closed subset of X has the perfect set property if and only if every analytic subset of X has the perfect set property” is equivalent to b > ω1 (hence, in particular, it is independent of ZFC). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement “For every space X, if every Γ subset of X has the… 

The onto mapping property of Sierpinski

Define (*) There exists $(\phi_n:\omega_1\to \omega_1:n<\omega)$ such that for every uncountable $I$ which is a subset of $\omega_1$ there exists $n$ such that $\phi_n$ maps $I$ onto $\omega_1$.

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