A solution to the L space problem

Abstract

In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space L \mathscr {L} is a subspace of T ω 1 {\mathbb {T}}^{\omega _1} where T \mathbb {T} is the unit circle. It is shown to have a number of properties which may be of additional interest. For instance it is shown that the closure in T ω 1 \mathbb {T}^{\omega _1} of any uncountable subset of L \mathscr {L} contains a canonical copy of T ω 1 \mathbb {T}^{\omega _1} . I will also show that there is a function f : [ ω 1 ] 2 → ω 1 f:[\omega _1]^2 \to \omega _1 such that if A , B ⊆ ω 1 A,B \subseteq \omega _1 are uncountable and ξ > ω 1 \xi > \omega _1 , then there are α > β \alpha > \beta in A A and B B respectively with f ( α , β ) = ξ f (\alpha ,\beta ) = \xi . Previously it was unknown whether such a function existed even if ω 1 \omega _1 was replaced by 2 2 . Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality ℵ 1 \aleph _1 . The results all stem from the analysis of oscillations of coherent sequences ⟨ e β : β > ω 1 ⟩ \langle e_\beta :\beta > \omega _1\rangle of finite-to-one functions. I expect that the methods presented will have other applications as well.