1. "U is a neighborhood of x" refers to "U is an open set containing x" (Munkres's Topology, etc)
2. "A is a neighborhood x" refers to "A contains an open set containing x" (wiki)
For 2, an open set in A containing x is an open subset of "int A (interior of A)", which is the largest open set contained in a neighborhood A.
Definition. A space X is locally path connected at a point p in X if every open set containing p contains a path connected open set containing p. The space X is locally path connected provided that it is locally path connected at each of its points.
The below lemma is basically equivalent to what Massey's book ask you to prove.
Lemma. A space X is locally path connected if and only if for each open subset O of X, each path component of O is an open set.
Assume X is locally path connected. Let O be an open set in X; let P be a path component of O; let x be a point of P. By the definition of a locally path connected, we can choose a path connected open set U of X such that . Since U is path connected, it must lie entirely in the path component P of O. Thus, P is open in X.
Assume a path component of an open set O is open. Given a point x of X and an open set O of x, let P be the path component of O containing x. Since P is an open set containing x and is contained in O, it satisfies the conditions for a locally path connected for X. Thus, X is locally path connected.
Not all path components are open. For instance, the path components of as a subspace of a standard topology on are not open. A topologist's sine curve is another example that the path components are not necessarily open.