We establish that, for any metrizable space X, if every closed separable subspace of X has the Baire property, then the space X has the Baire property. The same reflection result holds for spaces \(C_p(X)\) . It is also shown that all closed separable subspaces of \(C_p(X)\) are Čech-complete if and only if X is a functionally countable P-space. If \(\overline{A}\) is pseudocomplete for every countable set \(A\subset C_p(X)\) , then \(C_p(X)\) is pseudocomplete. To show that zero-dimensionality does not reflect in closed separable subspaces of metrizable spaces, under Jensen’s Axiom \(\diamondsuit \) , we give an example of a connected metrizable space M such that \(\overline{A}\) is countable and hence zero-dimensional for every countable set \(A\subset M\) .
