Locally Path Connected Definition Why are the following two statements equivalent for any topological space $X$?
1) $X$ is locally path connected (meaning, it has a basis of path connected sets).
2)  Every point of $X$ has a path connected neighborhood.
Is it simply that a path connected neighborhood is an open set in the subspace topology?

They aren't equivalent.  Indeed, any path-connected space satisfies (2), since you can take the neighborhood to just be $X$ itself.  But not every path-connected space is locally path-connected (see https://math.stackexchange.com/a/135483/86856, for instance).
