Compact Spaces with a $P$-base

In the paper, we investigate (scattered) compact spaces with a $P$-base for some poset $P$. More specifically, we prove that, under the assumption $\omega_1<\mathfrak{b}$, any compact space with an $\omega^\omega$-base is first-countable and any scattered compact space with an $\omega^\omega$-base is countable. These give positive solutions to Problems 8.6.9 and 8.7.7 in \cite{Banakh2019}. Using forcing, we also prove that in a model of $\omega_1<\mathfrak{b}$, there is a non-first-countable compact space with a $P$-base for some poset $P$ with calibre~$\omega_1$.