A topological space is totally disconnected if its only connected subsets are one-point sets.
Is it true that if \(X\) has the discrete topology, it is totally disconnected?
Is the converse true? Justify your answers.
Solution
1. Every discrete space is totally disconnected.
Let \(C\subseteq X\) contain at least two points and choose \(x\in C\). Since \(X\) is discrete, both \[ \{x\} \qquad\text{and}\qquad C\setminus\{x\} \] are open in the subspace \(C\). They are disjoint, nonempty, and cover \(C\), so \(C\) is disconnected. Hence every nonempty connected subset of \(X\) is a singleton.
2. The space \(\QQ\) with its usual subspace topology is not discrete.
Every open interval about a rational \(q\) contains rational points other than \(q\), so no singleton \(\{q\}\) is open in \(\QQ\).
3. The space \(\QQ\) is totally disconnected.
Let \(C\subseteq\QQ\) contain distinct points \(a<b\). Choose an irrational number \(r\) with \[ a<r<b. \] Then \[ C_-=C\cap(-\infty,r), \qquad C_+=C\cap(r,\infty) \] are disjoint nonempty sets open in the subspace \(C\). Because \(r\notin\QQ\), they cover \(C\). Thus every subset of \(\QQ\) containing at least two points is disconnected.
4. The converse is false.
By 2, \(\QQ\) is not discrete, while by 3 it is totally disconnected.