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 $\\emptyset$ 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.

单词	ClopenSubset
释义	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 $\\emptyset$ 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.
随便看	ring adjunction RingfiniteIntegralExtensionsAreModulefinite ring-finite integral extensions are module-finite RingHierarchy ring hierarchy RingHierarchyWithLinks ring hierarchy with links? RingHomomorphism ring homomorphism RingOfContinuousFunctions ring of continuous functions RingOfEndomorphisms ring of endomorphisms RingOfExponent ring of exponent RingOfSets ring of sets RingOfSintegers ring of S-integers RingsOfRationalNumbers rings of rational numbers RingsWhoseEveryModuleIsFree rings whose every module is free RingWithoutIrreducibles ring without irreducibles