clopen subset

A subset of a topological space $X$ is called clopen if it is bothopen and closed.

Theorem 1.

The clopen subsets form a Boolean algebra under the operation ofunion, intersection and complement. In other words:

• $X$ and $\mathrm{\varnothing }$ are clopen,

• the complement of a clopen set is clopen,

• finite unions and intersections of clopen sets are clopen.

Proof.

The first follows by the definition of a topology, the second bynoting that complements of open sets are closed, and vice versa,and the third by noting that this property holds for both openand closed sets.∎

One application of clopen sets is that they can be used todescribe connectness. In particular, a topological space isconnected if and only if its only clopen subsets are itself andthe empty set.

If a space has finitely many connected components then eachconnected component is clopen. This may not be the case if thereare infinitely many components, as the case of the rational numbersdemonstrates.