Examples of rare, meager and nonmeager sets in $\mathbb{R}$ Kreyszig Functional Analysis book presents the following definition.

I'm trying to get some examples.
(a) The cantor set $K$ is rare in $\mathbb{R}$ because it's closed and has empty interior so that $$\operatorname{int}\left(\overline{K}\right)=\operatorname{int}(K)=\varnothing.$$
(b) The set $\mathbb{Q}$ of rationals numbers is meager in $\mathbb{R}$ because it's enumerable so that we can write $$\mathbb{Q}=\{r_1,r_2,...,r_n,...\}=\bigcup_{i=1}^\infty{\{r_i}\},$$ where $\{r_i\}$ is rare for all $i\in\mathbb{N}$.
(c) I think that $(0,1)$ (like any other non-degenerate interval) is a nonmeager set in $\mathbb{R}$. My question is: how to prove it?.
Thanks.
 A: Your first two examples are spot on.  
Actually, so is the third.
The proof of this relies on the Baire Category Theorem, mentioned in the comments above.  

Baire Category Theorem: If a metric space $X\neq\varnothing$ is complete, it is nonmeager in itself.

To see this, recall that any non-degenerate open interval $(a,b)$ is homeomorphic to the real line $\mathbb{R}$.  As $\mathbb{R}$ is a complete metric space, it follows that $(a,b)$ is completely metrizable (of course, the usual metric is not complete on $(a,b)$, but some other metric — one derived from a homeomorphism from $\mathbb{R}$ — makes it complete).  Therefore $(a,b)$ is nonmeagre-in-itself by the Baire Category Theorem.
If $(a,b) = \bigcup_n Z_n$ where each $Z_n$ is nowhere dense in $\mathbb{R}$, it would follow that each $Z_n$ is also nowhere dense in the subspace $(a,b)$, which would then contradict the fact that $(a,b)$ is nonmeagre-in-itself.

A quick comment about your second example: It is good that you note that the singleton sets in $\mathbb{R}$ are nowhere dense.  For instance, $\mathbb{N}$ is countable, but not meagre-in-itself, because $\mathbb{N}$ is discrete (all subsets are open) and so the singleton sets are not nowhere dense in $\mathbb{N}$.  (Of course, $\mathbb{N}$ is meagre in $\mathbb{R}$.)
