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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asked Jul 30, 2010 at 18:30
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$\begingroup$ A G-delta set of such spaces is precisely a completely metrisable separable metric space. $\endgroup$– Henno BrandsmaCommented Jul 30, 2010 at 20:00
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5$\begingroup$ In fact, the equivalent question in more modern terms: does every connected Polish space have a proper closed connected subspace ? $\endgroup$– Henno BrandsmaCommented Jul 31, 2010 at 6:19
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$\begingroup$ Does this question have a known answer or is it an open problem? $\endgroup$– Garabed GulbenkianCommented Jul 31, 2010 at 18:19
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1$\begingroup$ This seems very closely related to mathoverflow.net/questions/25171/… $\endgroup$– BS.Commented Aug 3, 2010 at 14:10
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2$\begingroup$ @BS: Looks equivalent, in fact. This question should probably be closed as a duplicate. $\endgroup$– Nate EldredgeCommented Nov 19, 2010 at 0:20
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1 Answer
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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
answered Sep 9, 2010 at 19:54