Complement of rationals has empty interior

Question

This question refers to How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$

I want to prove that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I am trying to understand the accepted answer, but when it comes to "This shows that the complement of $\mathbb{Q}$ has empty interior, so the closure of $\mathbb{Q}$ is all of $\mathbb{R}$.", I get stuck.

What does it mean intuitively that a set has empty interior?

What I know is that a point is interior to a set when it is the center of some open ball inside that set. In that sense, for a set to have empty interior would mean that it has no interior points. Furthermore, the set of all its adherent points - a point $m$ is adherent to the set of rationals when there is a sequence of rational points $x_k$ such that $limx_k = m.$ Being so, the complement of rationals would have no interior points.

How does it follow from this that the closure of $\mathbb{Q}$ is $\mathbb{R}$?

Thanks in advance!

Answer 1

How does it follow from this that the closure of $\mathbb{Q}$ is $\mathbb{R}$?

Take $x \in \mathbb{R}$. Since $\mathbb{R}\ - \mathbb{Q}$ has empty interior, it follows that $x \notin int(\mathbb{R} - \mathbb{Q})$. It means that, for all $r > 0$, the ball $B(x,r)$ centred in $x$ with radius $r$ is not contained in $\mathbb{R}\ - \mathbb{Q}$. So, there is a $q \in \mathbb{Q}$ such that $q \in B(x,r)$. In other words, $x$ is in the closure of $\mathbb{Q}$. Since $x$ is arbritary, $\overline{\mathbb{Q}} = \mathbb{R}$.

Answer 2

This depends largely on the topology of the space that you're in. As copper.hat noted above, a set in a topological space has empty interior if it contains no open set. Recall that on $\mathbb{R}$ the open sets are generated by open intervals of the form $(a,b) = \{x \in \mathbb{R} \; | \; a <x < b\}$. So the complement of the rationals, which we can denote $\mathbb{R}\setminus \mathbb{Q}$, contains no interval. You can show this by demonstrating that a ball around any irraitonal point must contain a rational element.

Answer 3

a neighborhood of a point $x$ in $\mathbb{R}$ is an open interval, $(x- \epsilon _1 , x+ \epsilon _2 )$.

if $x$ is an interior point of a set $S \subseteq \mathbb{R}$, then x has a neighborhood of that flavor, which is contained in $S$.

if a set doesn't have any interior points, then, given any point in the set, you can't find an open interval $(x-\epsilon , x+\epsilon)$ which is contained in the set

in other words, if a subset of $\mathbb{R}$ has empty interior, then it doesn't have any subsets which are intervals

Answer 4

In any topological space $X$, for any subset $A$ of $X$, the interior of $A$ is the union of all open subsets contained in $A$, and the closure of $A$ is the intersection of all closed subsets that contain $A$. This is not the definition of interior and closure, but these characterizations follow easily from the definitions. (And each of the two propositions follows from the other, too, by taking complements). In particular, it follows that $A$ is dense in $X$ if and only if the complement of $A$ has empty interior. (The first condition means that $X$ is the only closed subset containing $A$; by looking at complements, this is the same as: the only open subset contained in the complement of $A$ is the empty set.)

Note that all this holds in any topological space, not only in metric spaces. There are no hypotheses required on $X$ or on $A$.

Answer 5

Every real number can be approximated arbitrarily closely by rational numbers; this is the basis for decimal notation. Now for an open set contained in the complement of $\Bbb Q$, its elements by definition cannot be approximated arbitrarily closely by rational numbers. So such a set is empty QED.