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

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 ${\mathbb{N}}^{\mathbb{N}}\mathrm{/}\mathcal{U}$.