Let each $X_i$ be a finite metric space. Define $X = \prod_{i\geq 1} X_i$ to be the product space equipped with the uniform metric, ie.
$$\rho_{\infty}(\mathbf{x}, \mathbf{y}) = \sup_{i\geq 1} d(x_i,y_i)$$
Show that $X$ contains no isolated points.
This question was given to me in an analysis course, but I have a feeling that it is false. Consider each $X_i = \{0,1\}$ with $d(0,1)=d(1,0)=1$. This is a metric space, of course, with $d(0,0)=d(1,1)=0$, $d(1,0) = d(0,1) = 1$, and $d(0,1) \leq d(0,0) + d(1,0)$ and vice versa.
If that is the case, however, then take $x = (0,0,\dots)$ and consider the ball $B_{\frac{1}{2}}(x) = \{x\}$. Clearly this is a contradiction.
Am I correct? Is the question presented in its current form false?
