I have kind of a simple point set topology question. If I am in ℝ2 and I have a connected open set, call it O, then is it true that all points on the boundary ∂O are limit points of O? I guess I'm stuck envisioning as O as, at least homeomorphic, to an open disk of radius epsilon. So it seems obvious that any points on the boundary would be limit points. But is that true in general?