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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?

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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.

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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:

3.3.4 Definition. Let $(X,\tau)$ be a topological space. Then it is said to be $\color{red}{\text{connected}}$ if the only clopen subsets of $X$ are $X$ and $\varnothing$.

In other words, $X$ is not connected if and only if it contains a clopen set other than $X$ and $\varnothing$.

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