Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other words (equivalently), let $\ C\ $ be a complement in $\ H\ $ of a $\sigma$-compact set (where a $\sigma$-compact set is simply a countable union of compacta in $\ H$).
QUESTION: Is $C$ necessarily connected and locally connected?
The answer is clearly "NO", if $H$ is a finite dimensional Euclidean space.