Clopen set

In topology, a clopen set (or closed-open set) in a topological space is a set which is both open and closed.

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 intervals [0,1] and [2,3]. 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 √2 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.)

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