In trying to prove that binary space (a homeomorphic space to the better known Cantor Set) is totally disconnected, what traits of the space do I need? Is binary space point-wise open (that would certainly be sufficient)?
As a quick reminder $B$ is the set of infinite sequences of $0$'s and $1$'s where for $x=(x_1,x_2,x_3,\cdots)$ and $y=(y_1,y_2,y_3,\cdots)$:
$$ d(x,y) = \begin{cases} \frac 1 n, & \text{n=min}[i|x_i\neq y_i] \\ 0, & x_i=y_i \end{cases} $$
I think the most general fact that applies here (and I can think of) is that every ultrametric space is totally disconnected, as balls are closed.
An ultrametric space is a metric space satisfying strong triangle inequality, i.e. $d(x,z)\leq \max(d(x,y),d(y,z))$. It is routine to check that it is satisfied by the metric you cited, and it is not hard to see that it implies that balls are closed.