TITLE: How thinly connected can a closed subset of Hilbert space be?
QUESTION [19 upvotes]: Let H be a separable (and infinite-dimensional) Hilbert space. Is it known whether there exists an infinite
subset C of H with the following properties.? (1) C is connected and closed in H. (2) No infinite proper 
subset of C is both connected and closed in H.
Perhaps the answer depends upon whether or not the Axiom of Choice is assumed to hold. In any case, no 
finite-dimensional Euclidean space-because it is locally compact-can contain a subset with these properties.

REPLY [6 votes]: Problem has now been solved.
There is a Polish space that is widely-connected. 
A space is widely-connected if it is connected and  each connected subset with more than one point is dense (so every connected subset is “wide”).  In the posts and comments above there are several equivalent definitions. 

Surely continuum theorists have thought about this question, haven't they?

Yes, they have. It appeared in various problem books which is how I became aware of it. The earliest mention I could find was actually by Paul Erdős, according to Mary Ellen Rudin.  It probably goes back to the 1940's, when Erdős was publishing on connected sets and dimension theory.