Ultrafilter characters and local $\pi$-bases

Abstract

We completely resolve Question~5.3 of Kania's paper \cite{Kania} and establish a more general realization theorem. We show that for infinite cardinals $\mu ⩽\nu <\kappa <\lambda$, a space with \[ \pc(X)=μ,\qquad\pcl(X)=ν,\qquad \pcc(X)=κ,\qquad\pcd(X)=λ\] exists if and only if $\kappa ⩽2^{\mu }$. The witnesses may be zero-dimensional and Tychonoff spaces, of the least possible cardinality $\lambda$. 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 $\pi$-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 $\pi$-bases for every finite $r$, centered value ${\omega }_1$, and any prescribed decreasing value greater than ${\omega }_1$.