I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].
Are there other examples?
UPDATE:
First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).
The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather, people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets": any space with this property is countable! Thus we reach a sad conclusion: looking at closed subsets of infinite deficiency does not give any new examples of spaces that always separate the Hilbert cube no matter how we embed them.
