Let H denote Hilbert space, the space of square-summable infinite sequences of real
numbers-which is infinite-dimensional and separable. Let S1,S2 denote subsets of H
such that a point p of H belongs to S1 or S2 according to whether the sum of the
abslolute values of the co-ordinates of p is convergent or divergent respectively.
Has the topology of these subsets of H been much investigated? It is clear that the
positive multiples of any point p of H remain in the same set (S1 or S2) that p itself
belongs to. Since the origin of H belongs to S1 it is clear that S1 is arcwise connected.
But is S2 also arcwise connected? I can see no easy way to prove this. 
Each point of H is a limit point of S1 and also a limit point of S2. Hence neither of
these sets is either an open or a closed subset of H. Is it possible that S1 and S2 are
not even Borel subsets of H? I started thinking about these sets in the course of trying
to come up with a series of positive rational numbers whose convergence or divergence
might be undecidable in Peano's Arithmetic or even in ZFC.



Questions about the topological properties of certain subsets of Hilbert space.
The unit ball of $\ell\_1$ is weakly compact in $\ell\_2$ and closed in $\ell\_2$, so $S1$ is $F\_\sigma$ and $S2$ is $G\_\delta$. 


ADDED 10/22/11: $S2$ is also arcwise connected. Give $x\_i$ in $S2$ for $i=1,2$ you can 
choose a partition $A \cup B$ of the natural numbers so that $1\_A x\_i$ and $1\_B x\_i$ are in $S2$ for $i=1,2$. The straight line paths from $x\_1$ to $1\_A x\_1 +1\_Bx\_2$ and from $1\_A x\_1 +1\_Bx\_2$ to $x\_2$ stay in $S2$.


