Connected space

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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.

As an example of a space that is not connected, one can delete an infinite line from the plane. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint disks in two-dimensional Euclidean space.

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Formal definition

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors specifically exclude the empty set with its unique topology as a connected space, but this encyclopedia does not follow that practice.

For a topological space X the following conditions are equivalent:

Connected components

The maximal connected subsets of a nonempty topological space are called the connected components of the space. The components of any topological space X form a partition of X: they are disjoint, nonempty, and their union is the whole space. (For an empty topological space this partition consists of zero connected components.) Every component is a closed subset of the original space. The components in general need not be open: for instance the components of the rational numbers, as subspace of the real numbers, are the one-point sets.

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