A solution to the L space problem and related ZFC constructions
Justin Tatch Moore
Abstract
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 [omega1]2 to omega1 such that if A,B, subsets of omega1, are uncountable and x omega1, 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 omega1 was replaced by 2. Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality aleph1. 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 ei : i < omega1 of finite-to-one functions. I expect that the methods presented will have other applications as well.
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