On cardinal sequences of length < omega3

TL;DR

This paper establishes the consistency of certain cardinal sequences of length less than omega_3 with GCH, using forcing to construct LCS spaces with prescribed sequences under specific set-theoretic assumptions.

Contribution

It proves the relative consistency of complex cardinal sequences of length less than omega_3 with GCH, expanding the understanding of LCS spaces in set theory.

Findings

Consistency of cardinal sequences of length < omega_3 under GCH.

Construction of LCS spaces with prescribed sequences.

For every uncountable cardinal, existence of LCS spaces with specific sequences.

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 $\eta < \om_3$ and every sequence $f = \langle \ka_{\al} : \al < \eta \rangle$ of infinite cardinals with $\ka_{\al}\leq \la$ for $\al < \eta$ 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 $\mbox{CS}(Z) = \langle \omega \rangle_{\alpha}\concat \langle \lambda \rangle_{\beta}$.