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

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