Ultrafilter characters and local π-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 μleqslantν<κ<λ, a space with pc(X)=μ, pcl(X)=ν, pcc(X)=κ, pcd(X)=λ exists if and only if κleqslant2μ. 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 ω1, and any prescribed decreasing value greater than ω1.
Related
BibTeX
@misc{kcakol2026ultrafilter,
title = {{Ultrafilter characters and local $π$-bases}},
author = {Jerzy Kcakol},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12612},
eprint = {2610.12612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12612},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ultrafilter characters and local π-bases
AU - Jerzy Kcakol
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12612
UR - https://arxiv.org/abs/2610.12612
AB - 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 μleqslantν<κ<λ, a space with pc(X)=μ, pcl(X)=ν, pcc(X)=κ, pcd(X)=λ exists if and only if κleqslant2^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.
ER -