Scheepers’ conjecture and the Scheepers diagram

Peng, Yinhe

doi:10.1090/tran/8787

Scheepers’ conjecture and the Scheepers diagram
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by Yinhe Peng;

Trans. Amer. Math. Soc. 376 (2023), 1199-1229

DOI: https://doi.org/10.1090/tran/8787

Published electronically: November 16, 2022

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Abstract:

Inductively approaching subsets by almost finite sets, we refute Scheepers’ conjecture under CH. More precisely, we prove the following.

Assuming CH, there is a subset of reals $X$ such that $C_p(X)$ has property ($\alpha _2$) and $X$ does not satisfy $S_1(\Gamma , \Gamma )$.

Applying the idea of approaching subsets by almost finite sets and using an analogous approaching, we complete the Scheepers Diagram.

$U_{fin}(\Gamma , \Gamma )$ implies $S_{fin}(\Gamma , \Omega )$.
$U_{fin}(\Gamma , \Omega )$ does not imply $S_{fin}(\Gamma , \Omega )$. More precisely, assuming CH, there is a subset of reals $X$ satisfying $U_{fin}(\Gamma , \Omega )$ such that $X$ does not satisfy $S_{fin}(\Gamma , \Omega )$.

These results solve three longstanding and major problems in selection principles.

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