A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some partially ordered set (poset) $P$ if there exists base $(U_p[x])_{p\in P}$ $x$ such that $U_p[x]\subseteq U_{p’}[x]$ for all $p\geq p’$ in $P$. We prove compact countable, hence metrizable, it has countable scattered height and $\mathcal {K}(M)$-base separable metric $M$. Banakh [Dissertationes Math...