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 \[ πχ(X)=πχ_{\mathrm l}(X)=ω<πχ_{\mathrm c}(X)=ω_1 <πχ_{\mathrm d}(X)=ω_2. \] 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 $\mathfrak p$, with a matrix filter that gives the exact values $ω_1$ and $ω_2$.