Wikipedia / Clopen set
Wikipedia / Clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name.
Examples
the empty set are both clopen.
which consists of the union of the two open intervals The topology is inherited as the subspace topology from the ordinary topology on the real line 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.
be an infinite set under the discrete metric 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.
of all rational numbers of all positive rational numbers whose square is )
Properties
is connected itself.
A set is clopen if and only if its boundary is empty.
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If all connected components is locally connected if and only if it is a union of connected components.
is discrete if and only if all of its subsets are clopen.
Using the union and intersection form a Boolean algebra. Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
See also
Notes