In Locally Path Connected?

A topological space is called locally path-connected if it has a basis of path-connected neighbourhoods. In other words, if for every point x and neighbourhood V∋x, there exists a path-connected neighbourhood U⊂V that contains x.

Does locally path connected implies locally connected?

is locally path connected, thus locally connected; it is also connected. More generally, every locally convex topological vector space is locally connected, since each point has a local base of convex (and hence connected) neighborhoods.

Does path connected imply locally path connected?

A locally path-connected space is path-connected if and only if it is connected. The closure of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected. The connected components of a locally connected space are also open.