Is a set with a single element like {0} dense in itself? Self explanatory really. Couldn't find an answer. I know that there can be sets with a highest and lowest element that are dense in themselves for example 
$\mathbb{Q}\cap[0,1]$
but I'm not sure about sets with just a single element.
 A: A subset $X\subset A$ of a topological space $A$ is dense in itself if $X$ contains no isolated points.

An isolated point is any point $x\in X$ such that $\{x\}$ is open in $X$ (with the subspace topology).
So following this definition, if $X$ is a singleton, then $X$ is not dense in itself, since $X$ is open in $X$.
A: Thanks to Vsotvep to pointing out my mistake.
I'll let this (modified) answer stand because maybe sombody else does the same mistake as I with an unknown definition that reads dangerously close to a more well known one.
If $X$ is a topological space and A subset of X, then Wikipedia gives a dense set definition of A is dense in X:

Formally, a subset A of a topological space X is dense in X if for any
  point x in X, any neighborhood of x contains at least one point from A
  (i.e., A has non-empty intersection with every non-empty open subset
  of X). 
Equivalently, A is dense in X if and only if the only closed
  subset of X containing A is X itself. This can also be expressed by
  saying that the closure of A is X, or that the interior of the
  complement of A is empty.

So looking at both equivalent definitions it should be clear that if $A=X=\{0\}$, then A is dense in X.
But Wikipedia also gives dense in itself  definition for some subset $A$ of a given topological space $X$:

In mathematics, a subset A of a topological space is said to be dense-in-itself if A contains no isolated points. 

Using that definition, which is the one appropriate one for this question, as Vsotvep showed in their answer, $A$ is not dense in itself. 
