Consistency of distinct selection principles for tau-covers

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Let XX be a topological space. Write {\sf S}_1(\{\mathcal O_n\}_{n\in\mathbb{N}),\mathrm{T}) for the property that, from every sequence of covers On\mathcal O_n of the relevant type, one can select a single member from each cover so that the selected family belongs to T\mathrm{T}. Write S1(Ω,T){\sf S}_1(\Omega,\mathrm{T}) for the corresponding selection property for sequences of open ω\omega-covers, where T\mathrm{T} is the collection of open τ\tau-covers. Consistency conjecture for tau-covers. It is consistent that

S1({On}n∈N,T)≠S1(Ω,T).{\sf S}_1(\{\mathcal O_n\}_{n\in\mathbb{N}},\mathrm{T})\neq{\sf S}_1(\Omega,\mathrm{T}).

The paper notes that S1(Ω,T){\sf S}_1(\Omega,\mathrm{T}) implies Rothberger's property S1(O,O){\sf S}_1(\mathcal O,\mathcal O), so consistency of Borel's conjecture prevents replacing “consistent” by “provable.” The claim is presented only as a guess, and no resolution of the consistency statement is supplied.

References

Primary source

Boaz Tsaban, “Strong gamma-sets and other singular spaces”, arXiv:math/0208057 (2007).

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