What is $Int( \mathbb {Q}^n)$ , $Ext( \mathbb {Q}^n)$ , $Bdy( \mathbb {Q}^n)$?

Question

What is $Int( \mathbb {Q}^n)$ , $Ext( \mathbb {Q}^n)$ , $Bdy( \mathbb {Q}^n)$?

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By looking at solutions, I know that:

$Int( \mathbb {Q}^n)=\emptyset $

$Ext( \mathbb {Q}^n)=\emptyset $

$Bdy( \mathbb {Q}^n)= \mathbb{R^n}$

I am looking for an explination.

Answer 1

$\newcommand{\Int}{\operatorname{Int}}\newcommand{\Ext}{\operatorname{Ext}}\newcommand{\Bou}{\operatorname{Bdy}}$ $\Int \mathbb{Q}^n = \emptyset$ since $\mathbb Q^n$ does not contain any open balls and hence any open set.

$\Ext \mathbb{Q}^n = \emptyset$ since $\mathbb Q^n$ is dense in $\mathbb R^n$, and hence the interior of its complement is empty.

$\Bou \mathbb Q^n = \mathbb{R}^n$, since the closure of $\mathbb{Q}^n$ is $\mathbb{R}^n$, and boundary is the difference between the closure and the interior.