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 $\mathscr {L}$ 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.