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locally connected (Definition)

A topological space $ X$ is locally connected at a point $ x \in X$ if every neighborhood $ U$ of $ x$ contains a connected neighborhood $ V$ of $ x$. The space $ X$ is locally connected if it is locally connected at every point $ x \in X$.

A topological space $ X$ is locally path connected at a point $ x \in X$ if every neighborhood $ U$ of $ x$ contains a path connected neighborhood $ V$ of $ x$. The space $ X$ is locally path connected if it is locally path connected at every point $ x \in X$.



"locally connected" is owned by djao.
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See Also: connected space, path, semilocally simply connected

Also defines:  locally path connected

Attachments:
a connected and locally path connected space is path connected (Theorem) by Mathprof
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Cross-references: path connected, connected, contains, neighborhood, point, topological space
There are 15 references to this entry.

This is version 2 of locally connected, born on 2002-05-17, modified 2003-10-06.
Object id is 2912, canonical name is LocallyConnected.
Accessed 5263 times total.

Classification:
AMS MSC: 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )
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