On cardinal sequences of length < omega3

Abstract

We prove the following consistency result for cardinal sequences of length $< \om_3$: if GCH holds and $\la \geq \om_2$ is a regular cardinal, then in some cardinal-preserving generic extension $2^{\om} = \la$ and for every ordinal $η< \om_3$ and every sequence $f = \langle \ka_{\al} : \al < η\rangle$ of infinite cardinals with $\ka_{\al}\leq \la$ for $\al < η$ and $\ka_{\al} = \om$ if $\mbox{cf}(\al) = \om_2$, we have that $f$ is the cardinal sequence of some LCS space. Also, we prove that for every specific uncountable cardinal $\lambda$ it is relatively consistent with ZFC that for every $\al,\be < \om_3$ with $\mbox{cf}(\al) < \om_2$ there is an LCS space $Z$ such that $\mbox{CS}(Z) = \langle ω\rangle_α\concat \langle λ\rangle_β$.