Connected Set

A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.

Let X be a topological space. A connected set in X is a set A subset= X which cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set A. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. The space X is a connected topological space if it is a connected subset of itself.

The real numbers are a connected set, as are any open or closed interval of real numbers. The (real or complex) plane is connected, as is any open or closed disc or any annulus in the plane. The topologist's sine curve is a connected subset of the plane. An example of a subset of the plane that is not connected is given by

Geometrically, the set B is the union of two open disks of radius one whose boundaries are tangent at the number 1.

See also

Closed Set, Connected Space, Empty Set, Open Set, Set, Set Closure, Simply Connected, Subset

Portions of this entry contributed by Matt Insall (author's link)

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References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 3, 1999.

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Connected Set

Cite this as:

Insall, Matt and Weisstein, Eric W. "Connected Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConnectedSet.html

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