Noetherian Properties, Large Cardinals, and Independence Around $\aleph_{\omega}$

Abstract

A base of a topological space is called {\em Noetherian } iff it does not contain an infinite strictly $\subseteq$-increasing chain. We show that minimal cardinality of a regular spaces without a Noetherian base is the first strongly inaccessible cardinal, answering a question from the 1980s. We also study the {\em Noetherian type} of a topological space $X$, denoted by $Nt(X)$, defined as the least cardinal $\kappa$ such that $X$ has a base $\mathcal B$ with $|\{B'\in \mathcal B: B\subset B'\}|<\kappa$ for each $B\in \mathcal B$. The behavior of the Noetherian type under the $G_\delta$-modification was investigated by Milovich and Spadaro. A central question, posed by them, is whether the Noetherian type of the $G_{\delta}$-modification of the space $D(2)^{\aleph_\omega}$ is $\omega_1$. This statement, denoted (Nt), is known to be independent of ZFC + GCH: it holds under ``GCH + $\square_{\aleph_\omega}$'', but fails under ``GCH + $(\aleph_{\omega+1}, \aleph_\omega)\to (\aleph_1, \aleph_0)$''. We place this phenomenon in a broader context by identifying similar independence phenomena for several topological and combinatorial principles. These include: (wFN) the weak Freese-Nation property of $[\aleph_{\omega}]^{\omega}$; (SAT) the existence of a saturated MAD family in $[\aleph_{\omega}]^{\omega}$; (HnT) the existence of an ${\omega}$-homogeneous, but not ${\omega}$-transitive permutation group on $\aleph_{\omega}$; and (SPL) the existence of a countably compact, locally countable, and ${\omega}$-fair regular space of cardinality $\aleph_{\omega+1}$. Assuming GCH, we analyze the logical relationships between these principles and show, for example, that SPL implies wFN, which in turn implies both SAT, HnT and Nt, while SAT does not imply Nt.