Ultrafilter characters and local $π$-bases

Abstract

We completely resolve Question~5.3 of Kania&#39;s paper \cite{Kania} and establish a more general realization theorem. We show that for infinite cardinals $μ\leqslantν<κ<λ$, a space with \[ \pc(X)=μ,\qquad\pcl(X)=ν,\qquad \pcc(X)=κ,\qquad\pcd(X)=λ\] exists if and only if $κ\leqslant2^μ$. The witnesses may be zero-dimensional and Tychonoff spaces, of the least possible cardinality $λ$. They may have the least possible number of non-isolated points, or alternatively be crowded. The main ingredient is an exact computation of the centered local $π$-character of star filter spaces in terms of ultrafilter character, obtained from a finite selection lemma for filter pushforwards. We also construct a single space with countable $r$-linked local $π$-bases for every finite $r$, centered value $ω_1$, and any prescribed decreasing value greater than $ω_1$.