Is the closure of a meager set meager?

Question

How to show that the closure of a meager set is meager?

I tried like this: Suppose that it is not meager then $cl(A)$, where $A$ is a meager set in a metric space $(X,d)$ contains an interior point and so an open ball around that point contained in $cl(A)$.

And this is where I am stuck, thanks.

Answer 1

As André Nicolas noted, this is not true: the set of rationals $\mathbb{Q}$ (with $\overline{\mathbb{Q}} = \mathbb{R}$) is a counterexample.