Examples
In any topological space X, the empty set and the whole space X are both clopen.
Now consider the space X which consists of the union of the two open intervals (0,1) and (2,3) of R. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set (0,1) is clopen, as is the set (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.
As a less trivial example, consider the space Q of all rational numbers with their ordinary topology, and the set A of all positive rational numbers whose square is bigger than 2. Using the fact that is not in Q, one can show quite easily that A is a clopen subset of Q. (Note also that A is not a clopen subset of the real line R; it is neither open nor closed in R.)
Read more about this topic: Clopen Set
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