A Measure Theoretic Proof of $\mathfrak p=\mathfrak t$

TL;DR

This paper provides a measure theoretic proof that the cardinal invariants  p and t are equal, addressing longstanding questions in set theory and analyzing the structure of filter-bases and ultrapowers.

Contribution

It offers a novel measure theoretic proof of  p= t, solving a longstanding problem and exploring the refinement of filter-bases to towers.

Findings

Confirmed  p= t using measure theory

Resolved the question of refining filter-bases to towers

Established a gap spectrum result for ultrapowers                    

Abstract

Rothberger's question of whether the two cardinals $\mathfrak p$ and $\mathfrak t$ are equal, posed back in 1948, was only answered fairly recently in the affirmative. Here we answer the more difficult progenitor question (posed in the same place) on when filter-bases can be refined to towers. Of equal import, our proof that $\mathfrak p=\mathfrak t$ addresses issues raised by Gowers concerning the "proof path" of the original solution. This is applied to obtain a gap spectrum result for the ultrapowers $\mathbb N^{\mathbb N}\,/\,{\mathcal U}$.