Totally disconnected space

Definition
A topological space is said to be totally disconnected if its connected components are one-point sets.

Examples
The rational numbers form a totally disconnected space. In fact, any irrational number gives a disconnection by partitioning the rational numbers into two open subsets -- the subset of numbers less than the given irrational and the subset of numbers greater than the given irrational.

Textbook references

 * , Page 152, Exercise 5 (definition introduced in exercise)