On metrizable subsets of hereditarily normal compact spaces - Junnila, Heikki; Tomoyasu, Kazuo (Topology Proceedings, 2021)
Let $X$ be a metrizable space which has a hereditarily normal $ω_1$-compactification. We show that $X$ is rim-separable and that if $X$ is also connected, then $w(X) ≤ ω_1$ and $X$ has a $σ$-point-finite base by sets with separable boundaries.