Connectedness and complexity in Polish spaces I was wondering: How complex can connected subsets of Polish spaces be?
Are there connected non-Borel subsets of a Polish space? Given $X$ Polish space (not totally disconnected), does it have proper analytic ( $\boldsymbol{\Sigma}_1^1(X)\setminus \boldsymbol{\Delta}_1^1(X)$ ) connected subsets?
Thanks!
 A: Well, for some Polish spaces, connected sets are very restricted--for instance, in $\mathbb{R}$ they are obviously all Borel.  But in general Polish spaces, they can pretty much be as bad as you want.  For instance, in $\mathbb{R}^2$, you can construct a connected set at any desired level of the projective hierarchy (including not projective at all!) as follows.  Start with a set $A\subset\mathbb{R}$ at the desired level, and then take the set $(\{0\}\times\mathbb{R})\cup (\mathbb{R}\times A)$.
Alternatively, given any collection of $\mathfrak{c}$ subsets of $\mathbb{R}^2$ of size $\mathfrak{c}$, by a transfinite recursion of length $\mathfrak{c}$ you can build a set $X\subseteq\mathbb{R}^2$ which intersects all of your sets but does not contain any of them.  Assuming your collection includes all uncountable closed sets, such an $X$ is automatically connected (see this answer).  So for instance, taking the collection of uncountably closed sets, this gives a connected subset of $\mathbb{R}^2$ without the perfect set property.  Or assuming CH (or just CH for projective sets), you can take the collection of all uncountable projective sets and build a connected subset of $\mathbb{R}^2$ that is not projective and does not even contain any uncountable projective set.
