Properties
- A topological space X is connected if and only if the only clopen sets are the empty set and X.
- A set is clopen if and only if its boundary is empty.
- Any clopen set is a union of (possibly infinitely many) connected components.
- If all connected components of X are open (for instance, if X has only finitely many components, or if X is locally connected), then a set is clopen in X if and only if it is a union of connected components.
- A topological space X is discrete if and only if all of its subsets are clopen.
- Using the union and intersection as operations, the clopen subsets of a given topological space X form a Boolean algebra. Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Read more about this topic: Clopen Set
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)