Consistency of distinct selection principles for tau-covers

From papers

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}nN,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.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Sources & referencesView supporting material

Primary source

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

Solutions 0

No solutions have been posted yet.