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A solution to the L space problem

Author: Justin Tatch Moore
Journal: J. Amer. Math. Soc. 19 (2006), 717-736
MSC (2000): Primary 54D20, 54D65, 03E02, 03E75; Secondary 54F15
Published electronically: December 21, 2005
MathSciNet review: 2220104
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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 $ A$ and $ B$ respectively with $ f (\alpha,\beta) = \xi$. Previously it was unknown whether such a function existed even if $ \omega_1$ was replaced by $ 2$. 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.

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Additional Information

Justin Tatch Moore
Affiliation: Department of Mathematics, Boise State University, Boise, Idaho 83725

Keywords: L space, negative partition relation, Tukey order, hereditarily Lindel\"of, non-separable, basis.
Received by editor(s): January 8, 2005
Published electronically: December 21, 2005
Additional Notes: The research presented in this paper was funded by NSF grant DMS–0401893.
Dedicated: This paper is dedicated to Stevo Todorcevic for teaching me how to traverse $𝜔_{1}$ and for his inspirational [23].
Article copyright: © Copyright 2005 American Mathematical Society