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 $\ H$).



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> **QUESTION:** Is $C$ necessarily connected and locally connected?
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The answer is clearly "NO", if $H$ is a finite dimensional Euclidean space.



A question about open subsets of Hilbert space whose complements are compact sets
General fact: if we remove a countable collection $K\_1,K\_2,\ldots$ of compact sets from an infinite-dimensional Banach space $X$, the remaining set $V$ is locally and globally path-connected (actually any open ball is path-connected).


Proof. If $K$ is a compact subset of $X$, and $x\in X$ is a point, then the set union $\cup\_{z\in K} xz$ of segments $xz$ is a compact (it is the image of the compact set $[0,1]\times K\subset \mathbb{R}\times X$ under continuous map $(t,z)\to x+t(z-x)$.) Let $x$, $y$ be two points in $V$. The set $\Omega$ of points $z$ for which the segment $zx$ or $zy$ contains a point from some $K\_i$ is a countable union of compact sets, thus has Baire first category. Therefore there exists a point $z\notin \Omega$ and $x$, $y$ may be joined by a path $xzy$.


