Is this wikipedia page on topological dimension correct? Wikipedia states:

We want the dimension of a point to be $0$, and a point has empty boundary, so we start with
${\displaystyle \operatorname {ind} (\varnothing )=\operatorname {Ind} (\varnothing )=-1}$
Then inductively, $ind(X)$ is the smallest $n$ such that, for every ${ x\in X}$ and every open set $U$ containing $x$, there is an open set $V$ containing $x$, such that the closure of $V$ is a subset of $U$, and the boundary of $V$ has small inductive dimension less than or equal to $n − 1$. (If $X$ is a Euclidean $n$-dimensional space, $V$ can be chosen to be an n-dimensional ball centered at $x$.)

Firstly, is it standard to say that the boundary of a single point is empty? In $\mathbb R$, I thougt the interior of a single point is empty and thus the boundary is equal to the point itself.
Secondly, the requirement that the "boundary of $V$ has small inductive dimension less than or equal to $n − 1$", suggests to me that the concept is ill defined, because for any single point $x$, any open set that contains $x$ (except the whole space) has a boundary that is nonempty.
 A: This definition assumes that when you define small inductive dimension of a point, you consider the induced topology on that point, so you consider the point as a topological space itself, a singleton. A singleton has only one topology, and it is the discrete, so the boundary of a point is indeed empty. 
More generally, the boundary of any finite set is empty, assuming the space we start with is at least T$_1$, since then the induced topology on a finite set is the discrete, and every set in the discrete topology has an empty boundary. This should answer both questions that you state. 
Yes, when we try to define the small inductive dimension of the boundary you may assume that: First, we consider the induced topology on the boundary (so we consider the boundary as a topological space on its own, kind of forgetting the space in which is may be a subset), and then Second, we apply the inductive definition on this subspace. 
I admit I was confused myself by the statement "a point has empty boundary" and my confusion cleared only after I read the answer by Theo Bendit, a link to which I have included above in a comment. But at hindsight, the statement is correct and "should" be obvious, given that it makes no reference to any larger space in which that point could sit. So it must be meant that the point is considered as a topological space itself (in which case, as noted above, it does have an empty boundary). 
