I was taking a look at this book. It says (p 102) that
If a topological space (X, T) is not connected, than X contains a subset U such that U isn't X, U isn't empty and U is clopen.
The book brings no proof of the fact. How does this imply?
This follows immediately from the fact that if $X$ is not connected, we can write
$$X = U \cup V$$
with $U$ and $V$ disjoint, not empty and both open in $X$. Since they are disjoint and their union is the whole space, we have $U = X- V$ and so $U$ is closed at the same time.
The answer depends on which definition of connectedness is being used. In this case it’s trivial, because the definition used in this book is as follows:
In other words, $X$ is not connected if and only if it contains a clopen set other than $X$ and $\varnothing$.