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Separated sets - Relation to connected spaces

Separated sets - Relation to connected spaces: Encyclopedia II - Separated sets - 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 < ...

Separated sets: Encyclopedia II - Separated sets - Relation to connected spaces

Separated sets - 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.

Other related archives

1/2, Connected space, Disjoint sets, Hausdorff space, Separation axiom, Topological distinguishability, Topology, closed, closure, complement, connected spaces, continuous function, distinct, empty set, if, intersection, intervals, mathematics, neighbourhood, open, open set, positive real number, preimage, real line, separable spaces, separated spaces, separation axioms, set theory, singleton sets, subset, subsets, topological space, topology, unit interval

Adapted from the Wikipedia article "Relation to connected spaces", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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