Are you confusing the subset of with as a metric space by itself? As a subset of , no subset of is open. But as a space in and of itself every subset of is open.

To reconcile this idea think of the ball centered at 1/4 with radius 1/25. Is that a subset of {1/4}? Is that not the definition of an open set? So {1/4} is an open set in the metric space .