On the intersection of non meager sets.

Question

Suppose that we have a topological space $X$ and a subspace $Z\subseteq X$ such that $Z$ is not meager in $X,$ if one has anhother subspace $Y$ which is not meager in itself, is it true that $Z\cap Y$ is not meager in $Y$?

Provided that $Z\cap Y$ is non empty.

Best regards.

Answer 1

Take $X = \mathbb{R}^2$ and let $Y = \mathbb{R} \times \{0\}$ be the $x$-axis. Let $Z = Y^c \cup \{(0,0)\}$.