A solution to the L space problem and related ZFC constructions

2005/01/28 by Justin Tatch Moore · 1 citation
Mathematics · #math.GN #math.LO #msc:54D20 #msc:54D65 #msc:03E02 #msc:03E75

paper · pdf · doi:10.1090/s0894-0347-05-00517-5

published as J. Amer. Math. Soc. 19 (2006), no. 3, 717--736 · 27 pages

arxiv created 2005/01/28 · arxiv updated 2013/10/08

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 e_i : i < omega₁ of finite-to-one functions. I expect that the methods presented will have other applications as well.

Cited by

• New results about Q and Δ-spaces