TITLE: Is a dense set always infinite?
QUESTION [6 upvotes]: Learning about dense sets the classical example is that of $\mathbb Q$, the rationals, in $\mathbb R$.  The same interpretation is valid for irrationals in $\mathbb R$. I was wondering if a dense set needs to be infinite, because this is what intuition would suggest. Moreover, are dense sets always countably infinite?

REPLY [20 votes]: To answer the general question of "are dense sets always infinite": no, because certainly, if $X$ is a finite topological space, then $X$ is dense in itself.
For another example, if $X$ is any set with the indiscrete topology, then every nonempty subset of $X$ is dense.
For yet another example, let $X = \mathbb{R}$ with the topology determined by the Kuratowski closure operator
$$\operatorname{cl}(S) = \begin{cases} S, & 0 \notin S; \\ \mathbb{R}, & 0 \in S.\end{cases}$$
Then $\{ 0 \}$ is dense in $X$, yet $X$ is $T_0$.  (In fact, this example can easily be modified to give a $T_0$ topological space of any desired cardinality such that some single point is dense.)
On the other hand, in a $T_1$ topological space, every finite subset is closed.  So, if a finite subset of a $T_1$ topological space $X$ is dense, then $X$ itself must be finite.  Or, for the contrapositive, if $X$ is an infinite $T_1$ topological space, then every dense subset of $X$ is infinite.

REPLY [4 votes]: In an infinite metric space any dense set must be infinite too: all finite sets are closed and so their closure ( the same finite set) cannot equal the whole space. One dense set is the whole space, so it need not be countable, unless the whole space is.