$P$-bases and topological groups

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. 538 (2019), p. 141] gives positive answer Problem 8.6.8. Let $A(X)$ be the free Abelian group on $X$. It shown $Y$ retract of $A(Y)$ $A(X/Y)$ $Q$-base, then $P\times Q$-base. Also closed subspace $P$-base, $P$-base. Fréchet-Urysohn with $M$ first-countable, metrizable. And poset calibre $(\omega _1, \omega )$ $G$ precompact subset G strictly angelic. Applications function spaces $C_p(X)$ $C_k(X)$ are discussed. also give an example Boolean character $\leq \mathfrak {d}$ subsets metrizable but doesn’t $\omega ^\omega$-base _1<\mathfrak {d}$. Gabriyelyan, Kakol, Liederman [Fund. 229 (2015), pp. 129–158] consistent negative 6.5.