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A solution to the L space problem

Authors:	Justin Tatch Moore

Institution:	Department of Mathematics, Boise State University, Boise, Idaho 83725

Abstract:	In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space $\mathscr{L}$ is a subspace of ${\mathbb{T}}^{\omega_1}$ where $\mathbb{T}$ 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 $\mathbb{T}^{\omega_1}$ of any uncountable subset of $\mathscr{L}$ contains a canonical copy of $\mathbb{T}^{\omega_1}$ . I will also show that there is a function $f:\omega_1]^2 \to \omega_1$ such that if $A,B \subseteq \omega_1$ are uncountable and $\xi < \omega_1$ , then there are $\alpha < \beta$ in and respectively with $f (\alpha,\beta) = \xi$ . Previously it was unknown whether such a function existed even if $\omega_1$ was replaced by . Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality $\aleph_1$ . The results all stem from the analysis of oscillations of coherent sequences $\langle e_\beta:\beta < \omega_1\rangle$ of finite-to-one functions. I expect that the methods presented will have other applications as well.

Keywords:	L space negative partition relation Tukey order hereditarily Lindel\"of non-separable basis

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