Reflecting topological properties in closed separable subspaces

Abstract 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 $$C_p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . It is also shown that all closed separable subspaces of $$C_p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are Čech-complete if and only if X is a functionally countable P -space. If $$\overline{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> is pseudocomplete for every countable set $$A\subset C_p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , then $$C_p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is pseudocomplete. To show that zero-dimensionality does not reflect in closed separable subspaces of metrizable spaces, under Jensen’s Axiom $$\diamondsuit $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>♢</mml:mo> </mml:math> , we give an example of a connected metrizable space M such that $$\overline{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> is countable and hence zero-dimensional for every countable set $$A\subset M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> .