# What characteristics of binary space are needed to prove that it is totally disconnected?

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 $y=(y_1,y_2,y_3,\cdots)$:

$$d(x,y) = \begin{cases} \frac 1 n, & \text{n=min}[i|x_i\neq y_i] \\ 0, & x_i=y_i \end{cases}$$

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.