Clopen set

Examples

In any topological space ${\displaystyle X,}$ the empty set and the whole space ${\displaystyle X}$ are both clopen.

Now consider the space ${\displaystyle X}$ which consists of the union of the two open intervals ${\displaystyle (0,1)}$ and ${\displaystyle (2,3)}$ of ${\displaystyle \mathbb{R}.}$ The topology on ${\displaystyle X}$ is inherited as the subspace topology from the ordinary topology on the real line ${\displaystyle \mathbb{R}.}$ In ${\displaystyle X,}$ the set ${\displaystyle (0,1)}$ is clopen, as is the set ${\displaystyle (2,3).}$ This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.

Now let ${\displaystyle X}$ be an infinite set under the discrete metric – that is, two points ${\displaystyle p,q\in X}$ have distance 1 if they're not the same point, and 0 otherwise. Under the resulting metric space, any singleton set is open; hence any set, being the union of single points, is open. Since any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen.

As a less trivial example, consider the space ${\displaystyle \mathbb{Q}}$ of all rational numbers with their ordinary topology, and the set ${\displaystyle A}$ of all positive rational numbers whose square is bigger than 2. Using the fact that ${\displaystyle \sqrt{2}}$ is not in ${\displaystyle \mathbb{Q},}$ one can show quite easily that ${\displaystyle A}$ is a clopen subset of ${\displaystyle \mathbb{Q}.}$ (${\displaystyle A}$ is not a clopen subset of the real line ${\displaystyle \mathbb{R}}$; it is neither open nor closed in ${\displaystyle \mathbb{R}.}$)