A solution to the L space problem

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.