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import FormalConjecturesUtilConjecture about cardinality of Lindelöf spaces
The conjecture asks for a Lindelöf space where all singletons are G_δ sets and which has cardinality > 𝔠.
This is Problem 1 in https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf
Selected Old Open Problems in General Topology by A. V. Arhangel’skii
open Cardinal
namespace CardinalityLindelof
Is there a Lindelöf space with singletons as Gδ sets with cardinality greater than the continuum?
@[category research open, AMS 54]
theorem HasGδSingletons.lindelof_card :
∃ (X : Type) (_ : TopologicalSpace X), HasGδSingletons X ∧ LindelofSpace X ∧ 𝔠 < #X := ⊢ ∃ X x, HasGδSingletons X ∧ LindelofSpace X ∧ 𝔠 < #X
All goals completed! 🐙
-- TODO: under additional axioms (consistent with ZFC), such a space exists.
end CardinalityLindelof