Is it true that every 1st category subset of a 2nd category space has empty interior?

Question

Let $X$ be a metric space. Are these conditions equivalent:

1. Each set of the 1. category in $X$ has empty interior;

2. $X$ is of the 2. category.

It is obvious that $1 \Rightarrow 2$. Is it true that $2 \Rightarrow 1$?

Answer 1

Let's take $X = [0,1] \cup \mathbb{Q}$, with the subspace topology induced by $\mathbb{R}$. $X$ is of second category in itself, but the set $(2,3)\cap X$ is both open in $X$ and of first category in $X$.