We prove that if $X$ is a space in which every subspace of cardinality at most $\aleph_1$ is metrizable, and $X$ has density $\aleph_1$, then $X$ has weight $\aleph_1$. We also extend a reflection theorem of Alan Dow by proving that if $X$ is a space in which every subspace of cardinality at most $\aleph_1$ is metrizable, and $X$ has a dense set, conditionally compact in $X$, then $X$ is metrizable. Known examples show that the compactness-like condition on $X$ cannot be weakened to pseudocompactness or feeble compactness. Some proofs use elementary submodels of ZFC.