nLab
locally path-connected space

A space is locally path-connected if it has a basis of path-connected neighbourhoods. In other words, if for every point x and neighbourhood Vx, there exists a path-connected neighbourhood UV that contains x.

A locally path-connected space is connected if and only if it is path-connected. In any case, the connected components of a locally path-connected space are the same as its path-connected components.

Revised on July 4, 2013 06:25:14 by Toby Bartels (141.0.8.190)