Examples 3.1:

(i) The intervals $[0, 1]$ and $(0, 1)$ on the real line are connected. The only connected subsets of the real line are intervals (including the empty set). Hence the only connected subsets of $\mathbb{Z}$ are singletons and the empty set.

(ii) Product of connected spaces are connected. Thus the cube $[0, 1]\times [0, 1]\times [0, 1]$ is connected.

We now state the most basic theorem on connectedness whose proof ought to be done in standard courses on general topology and will not be repeated here.