L-spaces and the P-ideal dichotomy

Abstract

We extend a theorem of Todorčević: Under the assumption ($\mathcal{K}$) (see Definition 1.11),

$$ \boxtimes \left\{ \begin{gathered} any regular space Z with countable tightness such that \hfill \\ Z^n is Lindel\ddot of for all n \in \omega has no L - subspace. \hfill \\ \end{gathered} \right. $$

We assume $\mathfrak{p}$ > ω ₁ and a weak form of Abraham and Todorčević’s P-ideal dichotomy instead and get the same conclusion. Then we show that $\mathfrak{p}$ > ω ₁ and the dichotomy principle for P-ideals that have at most ℵ₁ generators together with ⊠ do not imply that every Aronszajn tree is special, and hence do not imply (ie1-4). So we really extended the mentioned theorem.