A solution to the L space problem and related ZFC constructions

TL;DR

This paper constructs a ZFC example of a non-separable hereditarily Lindelof space (L space), introduces a special function with uncountable subsets, and proves the non-existence of a basis for uncountable regular Hausdorff spaces, refuting longstanding conjectures.

Contribution

It provides the first ZFC construction of an L space, introduces a novel uncountable function, and shows no basis exists for certain uncountable spaces, challenging previous assumptions.

Findings

Constructed a non-separable hereditarily Lindelof space in ZFC.

Established the existence of a special uncountable function with particular properties.

Proved there is no basis for uncountable regular Hausdorff spaces of size aleph_1.

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 [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.