  In this paper I will construct a non-separable hereditarily Lindelof space (L
space) without any additional axiomatic assumptions. I will also show that
there is a function f from [omega_1]^2 to omega_1 such that if A,B, subsets of
omega_1, are uncountable and x omega_1, then there are a < b in A and B
respectively with f(a,b) = x. 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.
Each of these results gives a strong refutation of a well known and
longstanding conjecture. The results all stem from the analysis of oscillations
of coherent sequences {e_i : i < omega_1} of finite-to-one functions. I expect
that the methods presented will have other applications as well.
