$G_{\delta }$ semifilters and ${\omega }^{\ast }$

The ultrafilters on the partial order $([\omega {]}^{\omega },{\subseteq }^{\ast })$ are the free ultrafilters on $\omega$, which constitute the space ${\omega }^{\ast }$, the Stone-Cech remainder of $\omega$. If $U$ is an upperset of this partial order (i.e., a semifilter), then the ultrafilters on $U$ correspond to closed subsets of ${\omega }^{\ast }$ via Stone duality. If, in addition, $U$ is sufficiently “simple” (more precisely, $G_{\delta }$ as a subset of $2^{\omega }$), we show that $U$ is similar to $[\omega {]}^{\omega }$ in several ways. First, ${\mathfrak{p}}_U={\mathfrak{t}}_U=\mathfrak{p}$ (this extends a result of Malliaris and Shelah). Second, if $\mathfrak{d} = \mathfrak{c}$thenthereareultrafilterson$U$thatarealso$P$-filters (this extends a result of Ketonen). Third, there are ultrafilters on $U$ that are weak $P$-filters (this extends a result of Kunen). By choosing appropriate $U$, these similarity theorems find applications in dynamics, algebra, and combinatorics. Most notably, we will prove that $({\omega }^{\ast },+)$ contains minimal left ideals that are also weak $P$-sets.