We say the metric space X is locally path connected (lpc) if all balls are path connected sets.
Suppose X is lpc and that E is an open and connected subset of X. Prove that E is path connected.
A set is open if every point is an interior point. A set is connected if it cannot be written as the union of two open, disjoint, non-empty sets. A set is path connected if given any two points x,y in the set, there exists a path between them.
The Attempt at a Solution
None really. My professor suggested that for a fixed x, I consider the set Y of all y that are path connected to x and show that this set is all of E by somehow showing that if it were not, I could disconnect the space using Y and Y complement.
I just don't see the direction that this problem could take. Is it long? Is it short but requires ingenuity? What should I try to do? Where am I trying to end up?
Help, it's hard!