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Let M be a finite-dimensional Euclidean space or an infinite-dimensional separable Banach space. An inner limiting subset of M is a countable intersection of open subsets of M-these sets are usually called "G-delta" 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?

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A G-delta set of such spaces is precisely a completely metrisable separable metric space. – Henno Brandsma Jul 30 '10 at 20:00
In fact, the equivalent question in more modern terms: does every connected Polish space have a proper closed connected subspace ? – Henno Brandsma Jul 31 '10 at 6:19
Does this question have a known answer or is it an open problem? – Garabed Gulbenkian Jul 31 '10 at 18:19
This seems very closely related to… – BS. Aug 3 '10 at 14:10
@BS: Looks equivalent, in fact. This question should probably be closed as a duplicate. – Nate Eldredge Nov 19 '10 at 0:20

This is only a partial answer.

If M has a subset that is somewhere locally path-connected, 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

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