Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. 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.

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.

Related Topics:
Topological space - Union - Disjoint - Nonempty - Open set - Subset - Subspace topology - Empty set

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For a topological space X the following conditions are equivalent:

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• X is connected.
• X cannot be divided into two disjoint nonempty closed sets (This follows since the complement of an open set is closed).
• The only sets which are both open and closed (clopen sets) are X and the empty set.
• The only sets with empty boundary are X and the empty set.
• X cannot be written as the union of two nonempty separated sets.

The maximal nonempty connected subsets of any topological space are called the connected components of the space.

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The components form a partition of the space (that is, they are disjoint and their union is the whole space).

Related Topics:
Partition - Disjoint

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Every component is a closed subset of the original space.

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The components in general need not be open: the components of the rational numbers, for instance, are the one-point sets.

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A space in which all components are one-point sets is called totally disconnected. A space X is totally disconnected iff, for any two elements x and y of X, there exist disjoint open neighborhoods U of x and V of y such that X is the union of U and V.

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