I am having difficulty with a problem in one of my textbooks. It gives three definitions
$1$. A set is called a closed set if its complement is open.
$2$.The closure of a set E is the intersection of all Closed sets C of which E is a subset.
$3$.If is a sets closure is equal to $\mathbb R^n$, then it is dense.
The question asks to show that the set of rational numbers are a dense subset of the reals. This is a second year university textbook. Here is what I have so far:
To show that the of set rational numbers is dense, it suffices to show that its closure is the set of Real Numbers. The closure of the set of rational numbers is defined to be the intersection of closed sets containing the rational numbers, and is thus the smallest closed set containing the rational numbers. A real number is defined to be the limit of a sequence of rational numbers, and therefore the smallest set that can contain all rational numbers (of which there exists a rational number between every two real numbers) as well as their boundary points (the irrational numbers) is the set of real numbers.
Best Answer
Let's say the closure of $\mathbb Q$ is a set ($\mathbb Q \subsetneq) X$ which is a proper subset of $\mathbb R$ ($X\subsetneq \mathbb R$).
Our aim is to show $X$ cannot be closed i.e. to show $\mathbb R -X$ is not open.
For any point $a\in \mathbb R- X$ for any open set $U$ containing a there exists a rational number in the open set $U$. Therefore any open set $U\not \subset R-X$ ( because $\mathbb R -X$ doesn't contain any rational point (as $\mathbb Q \subset X$)).
So, $\mathbb R -X$ is not open, and thus $X$ is not closed.
Therefore, any proper subset $X$ of $\mathbb R$ is not closed. So the smallest closed subset of $R$ containing $\mathbb Q$ is nothing but $\mathbb R$.
Conclusion- $\mathbb Q$ is dense in $\mathbb R$
Aside: About your question why the complement of empty set is $\mathbb R$ not $\mathbb C$.
When you talk of complement you always have an ambient space or an universal space to start off. So you will see the answer if you re-frame your question. Instead of questioning, "what is the complement of empty set?" one should question "what is the complement of empty set in $\mathbb R$. And the answer should be obviously $\mathbb R$.
If one asks "what is the complement of empty set in $\mathbb C$, then the answer will be $\mathbb C$.
I wish this helps.