Relation To Connected Spaces
Given a topological space X, it is sometimes useful to consider whether it is possible for a subset A to be separated from its complement. This is certainly true if A is either the empty set or the entire space X, but there may be other possibilities. A topological space X is connected if these are the only two possibilities. Conversely, if a nonempty subset A is separated from its own complement, and if the only subset of A to share this property is the empty set, then A is an open-connected component of X. (In the degenerate case where X is itself the empty set {}, authorities differ on whether {} is connected and whether {} is an open-connected component of itself.)
For more on connected spaces, see Connected space.
Read more about this topic: Separated Sets
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