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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..

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    Although they all have the same meaning, the definitions of interior, exterior, and boundary differs depending on where you first encounter them. Can you provide the definitions given in whatever text you are working out of? – 2017-01-18
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    (interior) There is an open rectangle $B$ such that $x\in B\subset A$ ---- (exterior) There is an open rectangle $B$ such that $x \in B \subset \mathbb {R}^n$ -> $A$ ---- (boundry) If $B$ is any open rectangle with $x \in B$ then $B$ cointains points of both $A$ and $\mathbb {R}^n$ -> $A$ – 2017-01-18
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    $\qquad\qquad\qquad\qquad\qquad\qquad$ $C=${x$\in$$\mathbb{R}^n$ $:$ each $x^i$ is rational}. $$\text{versus}$$ $$C=\{x \in \mathbb{R}^n : \text{each } x^i \text{ is rational}\}.$$ Can you finish cleaning this up based on what I've done? – 2017-01-18

1 Answers 1

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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,

$\operatorname{Bd}(\mathbb{Q}^n) = \operatorname{Cl}(\mathbb{Q}^n) - \operatorname{Int}(\mathbb{Q}^n) = \mathbb{R}^n$