In $\mathbb{R}$ with usual topology, a subset is connected if and only if it is an interval. $(0,1]$ is an interval (a connected set). $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ are all disconnected (totally disconnected) in $\mathbb{R}$ — they contain no intervals and every point is isolated from the rest in the subspace sense.