Is the interval $[a,b]$ connected with discrete metric?

Question

Is the interval $[a,b]$ connected with discrete metric?

My answer is no. But I got confused when I tried to prove it.

Answer 1

Using the discrete metric on a space $X$, all subsets of $X$ are both open and closed, and therefore any subset $A$ and its complement $A^c$ is a decomposition of $X$ into two open, disjoint, non-empty subsets (as long as $A \neq X, \varnothing$).

Answer 2

The sets $X = \{x \in [a,b] : d(x,a) < \frac 12\}$ and $Y = \{x \in [a,b] : d(x,a) > \frac 12\}$ are disjoint, both are nonempty and open in $[a,b]$, and their union is $[a,b]$.