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 $V\ni x$, there exists a path-connected neighbourhood $U\subset V$ 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.