TITLE: Smallest dense subset of $\mathbb{R}$
QUESTION [36 upvotes]: I am not sure if what I am looking for even makes sense (or) exists. Anyway I would be happy if someone can clear my confusion.
The set of real numbers $\mathbb{R}$ is obtained as completion of $\mathbb{Q}$. However, $\mathbb{Q}$ is not the only set which is dense in $\mathbb{R}$. $\mathbb{Q} \backslash \mathbb{Z}$ is also a dense subset of $\mathbb{R}$. I am wondering if it makes sense to talk of the "smallest" dense subset of $\mathbb{R}$. To phrase what I am looking for precisely and what I mean by smallest, I am looking for a dense set $A$ of $\mathbb{R}$ such that if $B$ is a proper subset of $A$ then $B$ is not dense in $\mathbb{R}$. I am able to "see" that the set $A$, I am looking for doesn't exist since any open interval consists of infinite rationals. But I am unable to precisely argue out to myself and convince why $A$ doesn't exist. Could some one throw more light on this?

REPLY [11 votes]: Let $X$ be a topological space which is separated and without isolated points: that is, for all $x \in X$, the singleton set $\{x\}$ is closed and not open.  Let $Y$ be a dense subset of $X$.  
CLAIM: For every nonempty open subset $U$ of $X$, $U \cap Y$ is infinite.  
Proof: First observe that $U$ is infinite: if not, removal of all but one of its points would make a singleton open set.  Then similarly, if $U \cap Y$ were finite, removal of all of its points would leave a nonempty open subset of $X$ which is disjoint from $Y$, contradicting denseness of $Y$.
It follows immediately from the claim that removing any finite number of points from $Y$ leaves us with a subset which still meets every nonempty open subset of $X$ so is still dense.  In particular, there is no minimal dense subset of $X$.  
In particular the argument applies to any metric space without isolated points, like 
$\mathbb{R}$.  It is a good exercise to check that both of the hypotheses imposed on $X$ are necessary.  Especially, in a non-separated space one may well have singleton dense subsets: these are called generic points and are ubiquitous (and useful) in algebraic geometry.