Let M be a finitedimensional Euclidean space or an infinitedimensional separable Banach space. An inner limiting subset of M is a countable intersection of open subsets of Mthese sets are usually called "Gdelta" sets nowadays. Does every infinite connected inner limiting subset of M contain an infinite connected proper subset H and a point p that is not a limit point of H?

This is only a partial answer. If M has a subset that is somewhere locally pathconnected, then yes. Otherwise, M is nowhere path connected, and this should put some interesting restrictions on M. Even in this case I do not see where M could nontrivially avoid all tuples (p,O) where p is a point and O a basis element of the space that does not contain p in its closure. Gerhard "Ask Me About System Design" Paseman, 2010.09.09 

