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.

Local connectedness

A topological space is said to be locally connected if it has a base of connected sets.

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It can be shown that a space X is locally connected if and only if every component of every open set of X is open.

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The topologist's sine curve shown above is an example of a connected space that is not locally connected.

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Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets.

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An open subset of a locally path-connected space is connected if and only if it is path-connected.

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This generalizes the earlier statement about Rn and Cn, each of which is locally path-connected.

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More generally, any topological manifold is locally path-connected.

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