Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path.
A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X.
An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with a closed annulus removed, as well as the union of two disjoint open disks in two-dimensional Euclidean space.
Formal definition
A topological space X is said to be disconnected if it is the union...