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 
be a topological
space. A connected set in 
is a set 
which 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. The space 
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
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Geometrically, the set 
is the union of two open disks of radius one whose boundaries are tangent at the
number 1.
