Question

prove that a discrete metric space is connected iff it has one element

Answer 1

think this is pretty obvious but that does not mean it is trivial to prove. if i recall "discrete" metric means
\[ d(x,y)= \left\{
\begin{array}{lr}
1 & : x=y\\
0 & : x \neq y
\end{array}
\right.
\]

Answer 2

what is precise definition of "connected"?

Answer 3

only open subsets both open and closed are M and empty set? is there something better?