# A question about connected inner limiting sets

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?

-
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 mathoverflow.net/questions/25171/… –  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
show 1 more comment