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Extent of the Vietoris power V(ω^ω)

Ascertain the extent e(V(ω^ω)), the supremum of cardinalities of closed discrete subspaces of the Vietoris power space V(ω^ω).

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Background

The authors show that the spread s(V(ωω)) equals 𝔠, but the exact extent e(V(ωω)) is left unresolved. Since extent ≥ spread for many spaces and e(V(ωω))=𝔠 would imply that V(ωω) is not Lindelöf and not normal, determining e(V(ωω)) would settle the other two open questions in the negative.

References

We use ? to indicate that we have not been able to verify whether the indicated space satisfies the indicated property, which we formalize as Question \ref{question:VBaire}. What is the extent of \mathsf V(\omega\omega)?

— An Adaptation of the Vietoris Topology for Ordered Compact Sets (2507.17936 - Caruvana et al., 23 Jul 2025) in Question 4.? (labelled Question \ref{question:VBaire}), Subsection “The Vietoris Power on Subsets of Naturals”