Simple question about discrete metric and openness. You may think this is silly question, but I'm really confused.
In discrete metric, every singleton is an open set.
And, the proof goes like this
$\forall x \in X$, by choosing $\epsilon < 1$, $N_\epsilon(x) \subset ${$x$}
However, if discrete metric is defined in $X$,
How can we use the radius less than 1 when we only have 0 and 1?  

A metric on $X$ is a mapping $d:X \times X \to [0,\infty[$, so you can take $d = 1/2$ even if no pairs with distance $1/2$ exist.
