Locally Connected

A topological space is locally connected at the point x if every neighborhood of x contains a connected open neighborhood. It is called locally connected if it is locally connected at every point.

A connected space need not be locally connected; counterexamples include the comb space and broom space. Conversely, a locally connected space need not be connected; an easy counterexample is the union of two disjoint open intervals of the real line.

See also

Connected im Kleinen, Locally Pathwise-Connected

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Locally Connected." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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