TITLE: Is a countable set $A$ dense in an uncountable set $B$ by analogy of $\mathbb{Q}$ being dense in $\mathbb{R}$?
QUESTION [7 upvotes]: Let $A$ be any countable set and $B$ any uncountable set with the same infinity as $\mathbb{R}$. Then we must have a one-to-one map $\phi:A\rightarrow\mathbb{Q}$. So, can we say by all this that $A$ is in fact dense in $B$, since it is well known that $\mathbb{Q}$ is dense in $\mathbb{R}$ ?

REPLY [6 votes]: Note that one can go to the almost opposite extreme as in Asaf's example above, and consider $\mathbb{R}$ with the discrete topology, so that all subsets are open.
Then the only dense set in this topological space is the entire set $\mathbb{R}$ itself.  (So, in particular, no countable subset is dense.)