Find the interior, exterior, and boundary of the sets Find the interior, exterior, and boundary of the sets:
$$\{x\in\mathbb{R}^n : |x| \leq 1\}$$
$$\{x\in\mathbb{R}^n : |x|=1\}$$
$$\{x\in\mathbb{R}^n : \text{ each } x^i \text{ is rational}\}.$$

I am new to this so bear with me. This is all I have so far...
I am assuming $|x|$ means norm. So $|x|= \sqrt{(x^1)^2+\cdots+(x^n)^2}$. If we let: 
$$A=\{x\in\mathbb{R}^n : |x| \leq 1\}$$
$$B=\{x\in\mathbb{R}^n : |x|=1\}$$
$$\{x\in\mathbb{R}^n : \text{ each } x^i \text{ is rational}\}.$$
Then $A\subseteq B \subset C$
Not sure where to go on from here..
 A: Let $S$ be a subset of a topological space $X$. Then $p$ is an interior point of $S$ if $p$ is contained in an open subset of $S$. 
If $S$ is a subset of a Euclidean space, then $p$ is an interior point of $S$ if there exists an open ball centered at $p$ which is completely contained in $S$.
$\forall q\in \mathbb{Q}^n$, and, $\forall \epsilon>0$
the ball 
$B_\epsilon(q)=\{x\in\mathbb{R}^n:|x-q|<\epsilon\}$ 
contain points with irrational coordinates, which are not in $\mathbb{Q}^n$ $\implies$ $q \notin$ $\operatorname{Int}(\mathbb{Q}^n)$.
Hence, 
$\operatorname{Int}(\mathbb{Q}^n) = \emptyset$
This also means that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$, i.e. $\mathbb{Q}^n$ has non-empty intersection with every non-empty open subset of $\mathbb{R}^n$.
Or, equivalently, the closure of $\mathbb{Q}^n$ is $\mathbb{R}^n$.
$\operatorname{Cl}(\mathbb{Q}^n) = \mathbb{R}^n$
The exterior of a subset $S$ of a topological space $X$ is defined as
$\operatorname{Ext}(S) = X - \operatorname{Cl}(S)$
Hence,
$\operatorname{Ext}(\mathbb{Q}^n) = \mathbb{R}^n - \operatorname{Cl}(\mathbb{Q}^n) = \emptyset$
The boundary of a subset $S$ of a topological space $X$ is defined as
$\operatorname{Bd}(S) = \operatorname{Cl}(S) - \operatorname{Int}(S)$
Hence,
$\operatorname{Bd}(\mathbb{Q}^n) = \operatorname{Cl}(\mathbb{Q}^n) - \operatorname{Int}(\mathbb{Q}^n) = \mathbb{R}^n$
