If \(X\) is a connected space if and only if \(X\) and the empty set are the only open and closed sets.
Solution
We interpret the printed sentence as the standard biconditional: \(X\) is connected if and only if the only subsets of \(X\) that are both open and closed are \(\varnothing\) and \(X\).
1. Suppose \(X\) is connected, and let \(C\subseteq X\) be both open and closed.Then \(C=\varnothing\) or \(C=X\).
Because \(C\) is closed, its complement \(X\setminus C\) is open. If both \(C\) and \(X\setminus C\) were nonempty, then \[ X=C\cup(X\setminus C) \] would be a union of two disjoint nonempty open sets, which would disconnect \(X\). Thus one of them is empty, giving \(C=\varnothing\) or \(C=X\).
2. Conversely, suppose \(\varnothing\) and \(X\) are the only clopen subsets of \(X\).Then \(X\) is connected.
Assume for contradiction that \(X\) is disconnected. Then there are disjoint nonempty open sets \(U,V\subseteq X\) with \[ X=U\cup V. \] Since \(X\setminus U=V\) is open, \(U\) is closed as well as open. Thus \(U\) is a nontrivial clopen subset of \(X\), contradicting the hypothesis.
3. Hence the two conditions are equivalent.
The forward implication is 1 and the reverse implication is 2.