Is $\mathbb R$ of second category in itself?

Question

I stumbled over the question

Is the closure of a set of first category also of first category?

and immediately thought of $\overline{\mathbb Q} = \mathbb R$ as a counterexample. Clearly, $\mathbb Q$ is of first category. But is $\mathbb R$? I highly assume it is not, but it doesn't seem to be as obvious as I thought. Any simple proofs?

Answer 1

$\mathbb{R}$ is a complete metric space with respect to the usual topology,so from Baire's theorem is of second category in itself.

A set of second category is a set which is not of first category,i.e it cannot be written as a countable union of nowhere dense sets.

So your counterexample is valid.

Also remember that Baire's theorem has this corollary:

Let $(X,d)$ be a complete metric space and $F_n$ a countable collection of closed sets ,such that $X=\bigcup_{n=1}^{\infty}F_n$

Then at least one of these closed sets $F_n$ has a nonempty interior.