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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
DOI: https://doi.org/10.1090/S0894-0347-05-00517-5
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
MR Author ID: 602643
Email: justin@math.boisestate.edu

Keywords: L space, negative partition relation, Tukey order, hereditarily Lindelöf, 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 $\omega _1$ and for his inspirational [23].
Article copyright: © Copyright 2005 American Mathematical Society