Three local π-characters
Abstract
We answer in ZFC a question of N. Carlson by constructing a zero-dimensional Tychonoff space X, with exactly one non-isolated point, for which every point admits a decreasing local π-base and πχ(X)=πχl(X)=ω<πχc(X)=ω1 <πχd(X)=ω2. A preliminary chain-refinement lemma shows that every inclusion-chain of subsets of a set has a reverse-well-ordered coinitial subchain of controlled cardinality. Thus the result is independent of whether a decreasing local π-base is understood as an inclusion-chain, a strictly decreasing transfinite sequence, or a weakly decreasing transfinite sequence. The construction combines a countable star filter, whose centred local π-character is at least the pseudointersection number p, with a matrix filter that gives the exact values ω1 and ω2.
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BibTeX
@misc{kania2026three,
title = {{Three local $π$-characters}},
author = {Tomasz Kania},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02379},
eprint = {2610.02379},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02379},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Three local π-characters
AU - Tomasz Kania
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02379
UR - https://arxiv.org/abs/2610.02379
AB - We answer in ZFC a question of N. Carlson by constructing a zero-dimensional Tychonoff space X, with exactly one non-isolated point, for which every point admits a decreasing local π-base and πχ(X)=πχ_l(X)=ω<πχ_c(X)=ω_1 <πχ_d(X)=ω_2. A preliminary chain-refinement lemma shows that every inclusion-chain of subsets of a set has a reverse-well-ordered coinitial subchain of controlled cardinality. Thus the result is independent of whether a decreasing local π-base is understood as an inclusion-chain, a strictly decreasing transfinite sequence, or a weakly decreasing transfinite sequence. The construction combines a countable star filter, whose centred local π-character is at least the pseudointersection number p, with a matrix filter that gives the exact values ω_1 and ω_2.
ER -